Morse Theory for Hamiltonian Systems

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CHAPMAN & HALUCRC Research Notes in Mathematics

Alberto Abbondandolo

Morse theory for Hamiltonian systems

CHAPMAN & HALUCRC

425


CHAPMAN & HAWCRC Research Notes in Mathematics Series Main Editors H. Brezis, Universitet de Paris R.G. Douglas, Texas A&M University A. Jeffrey, University of Newcastle upon Tyne (Founding Editor)

Editorial Board H. Atnann, University of Zurich R. Aris, University of Minnesota G.J. Barenblatt, University of Cambridge H. Begehr, Freie Universitd t Berlin P. Bullen, University of British Columbia R.J. Elliott, University of Alberta R.P. Gilbert, University of Delaware D. Jerison. Massachusetts Institute of Technology B. Lawson, Sense University of New York

B. Moodie. University ofAlberra S. Mori, Kyoto University L.E. Payne, Cornell University D.B. Pearson. University of Hull 1. Raebum. University of Newcastle, Australia G.F. Roach, University of Strathclyde 1. Stakgold, University of Delaware W.A. Strauss, Brown University

1. van der Hoek. University of Adelaide

at Story Brook

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Library of Congress Cataloging-in-Publication Data Abbondandolo. Alberto.

Morse theory for Hamiltonian systems / Alberto Abbondandolo. p. cm.- (Chapman & HaIVCRC research notes in mathematics series : 425) Includes bibliographical references and index. ISBN 1.58488-202-6 (alk, paper) 1. Morse theory. 2. Hamiltonian systems. 1. Title It. Series. QA331 .A23 2001 514'.74---dc2l

2001017240

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Contents Preface

ix

1 The Maslov index

.. . ... .. .. . ... . ..

The symplectic group . . . . The Maslov index in dimension 2 . . 1.2.1 The topology of Sp(2) . . . . . . 1.2.2 The rotation function on Sp(2) . . . . 1.2.3 The Maslov index for non-degenerate paths in Sp(2) 1.3 The Maslov index in dimension 2N . . 1.3.1 The Krein signature on Sp(2N) . . . . . 1.3.2 Normal forms of semi-simple symplectic matrices . . 1.3.3 The topology of Sp(2N) 1.3.4 The rotation function on Sp(2N) . . . . . 1.3.5 Decomposition of Sp(2N) . . . . 1.3.6 The Maslov index of non-degenerate paths in Sp(2N) 1.3.7 The Maslov index of degenerate paths in Sp(2N) . . 1.4 The Maslov index of a linear Hamiltonian system 1.4.1 Iteration formulas . 1.5 The Maslov index of an autonomous system 1.6 Some bibliography and further remarks 1.1 1.2

.. .. .. .. .

.. . ... .

. .... . .. . .... .. .. .. ..

. ..... . .. .... .

................ .. .. .. .. ..

.. . ... .

. .... .

.... . . ... . ... . ... . .. .. ... ..

.... .. .. ...

2 The relative Morse index 2.1

.

2.2 2.3

2.4 2.5 2.6

.... . .. .. . ... ..... . ..

Commensurable spaces and relative dimension 2.1.1 Further properties . . Fredholm pairs of subspaces . . . . . . Relative Morse index of quadratic forms 2.3.1 Further properties . . . . . . . . Relative Morse index of critical points . . . Finite dimensional reductions . Some bibliography and further remarks

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14

17 19

24 28 31 32 33 37 41

43 46 49 51

55

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57 58 62

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4 4 5 9 10 10

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1

43 .

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1

Contents

vi

3 Functional setting 3.1 3.2

.

.. ... . .. .. .. .. .. .. .. . ... . .. .. Linear Lagrangian systems .. .. .. .. .. 3.4.1 Comparison between the Morse index and the Maslov index .. .. .. .. .. .. Nonlinear Lagrangian systems .. .. . ... .

3.3 3.4

.

.

.

3.5 3.6

63

. .... ..

..

. . . . . Fractional Sobolev spaces . . . . . . . . . . . Linear Hamiltonian systems . 3.2.1 Comparison between the relative Morse index and the . . . . . . . . . . . Maslov index . . . . . . . . . Nonlinear Hamiltonian systems .

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Some bibliography and further remarks

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4 Superquadratic Hamiltonians 4.1

69 74 78

80 83 85

87

.. .. .. 87 .. .. .. 87 .. . ... . .. 90 4.1.2 A review of Morse theory 4.1.3 Linking theorems and Morse index estimates . ... 94 .. 4.1.4 Strongly indefinite functionals . .. ... 97 .. .. . ..... 100 Superquadratic Hamiltonians Abstract critical point theory .. . . . 4.1.1 The (PS) and (PS)* condition .

4.2

63 67

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4.2.1 Minimality of the period .. .. 4.3 A Birkhoff-Lewis type theorem 4.4 Some bibliography and further remarks

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108

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117

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5 Asymptotically linear systems 5.1

.. .... .

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119 .

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functionals,

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. . . . . . The Arnold conjectures . . . 6.1.1 Hamiltonian diffeomorphisms of the two-torus . 6.1.2 Non-degenerate fixed points on the torus .

6.1.3

132

.. ....... ..... 136

Systems with resonance at infinity 5.3.1 Abstract asymptotically quadratic degenerate at infinity . . . . . . . . . 5.3.2 Resonant systems . Some bibliography and further remarks .

6 The Arnold conjectures for symplectic fixed points

6.2

131

Growth of the number of periodic solutions of

..

6.1

120 120 126

Solutions of higher period ............... 129 autonomous systems ..

5.4

.

.. .. .. ..

Morse relations for autonomous systems 5.2.1

5.3

.

. . . Non-resonant systems . . 5.1.1 Abstract asymptotically quadratic functionals . 5.1.2 Morse relations for non-resonant systems .

5.1.3 5.2

.

136 147 151

153

.. .. .. ..

.

153 156 160

Degenerate fixed points on the torus ......... 163

The Arnold conjectures on the projective space

.

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165

Contents 6.3 6.4

vii

Periodic points on the torus ... . .... ... .. .. . 169 Some bibliography and further remarks ........... 171 .

.

Bibliography

173

Index

187

Preface This monograph deals with existence and multiplicity questions for periodic solutions of first order Hamiltonian systems p'

H(t, p, 9),

(p, q) E

R2N

Since the middle of the eighteenth century, it has been known that periodic solutions of such a system are extremals of the action functional elf (Z) =

Jpdq - fH(tz(f))dt

over closed curves in the phase space RZN. However, it was only in the last

twenty years that this functional was found to be a powerful instrument for proving the existence of periodic orbits. The main difficulty, which pre-

vented progress for a long time, is that periodic orbits do not minimize ey, they are just extremals. What is even worse is that, unlike the case of second order Lagrangian systems, these extremals have infinite Morse index and co-index: there are infinitely many directions where the functional decreases and infinitely many where it increases. Therefore, the standard critical point techniques, which detect extremals of a functional by looking at changes in the topology of the sublevels, cannot be applied directly: critical points with infinite Morse index and co-index are invisible to homotopy theory. The breakthrough dates to 1978, when Rabinowitz found a suit-able Hilbert space setting where em can be written as the sum of a non-degenerate quadratic form and a nonlinear term, with compact gradient. This fact allowed the development of new techniques in critical point theory, such as Benci and Rabinowitz's linking theorem, Amann's saddle point reduction, and Galerkin's reductions, and it provided the first results about existence of periodic orbits of Hamiltonian systems which are not close to complete integrability. Ix

x

Preface

In particular, the saddle point reduction and the Galerkin reduction enable the use of Morse theory, providing a fine instrument to prove multiplicity results. The aim of this book is to give a unified account of these techniques and to show selected applications. More specifically, here is a list of topics included in this volume.

Chapter 1. This chapter contains a detailed description of the Maslov index, a number which labels linear periodic Hamiltonian systems by looking at the windings made by their fundamental solution inside the symplectic group. The relevance of the Maslov index, as a number playing the role of the Morse index, was discovered by Conley and Zehnder in 1984. An extensive study of its properties has been carried on by Long and his collaborators. We start with the two-dimensional case, which allows a simple pictorial description. Then we define the Krein signature of the eigenvalues

of modulus one of a symplectic matrix and use this concept to build the rotation function on the symplectic group. This function plays the role of the determinant of the unitary part in the polar decomposition, but it has better invariance properties. The Maslov index is then defined for both non-degenerate and degenerate systems. Some of its iteration properties are studied, together with some useful formulas, as the one which expresses the Maslov index of an autonomous system in terms of the eigenvalues of its coefficients' matrix.

Chapter 2. Starting from the observation that the Hessian of the action functional at every curve consists of a compact perturbation of some fixed invertible self-adjoint operator, we introduce a suitable notion of relative Morse index, passing through the concept of relative dimension of commensurable subspaces. These notions provide us with the natural language to deal with strongly indefinite functionals and enable the simplification of many of the statements involving comparisons of indices. Since most of our existence results are obtained by applying critical point techniques to finite dimensional reductions of the action functional, we conclude this chapter by comparing the relative Morse index with the standard Morse index of reduced operators. Chapter S. After a brief introduction to fractional Sobolev spaces, we describe the functional setup for the variational theory of Hamiltonian systems. The main result is that the relative Morse index of a solution equals

the Maslov index of its linearization. Although in this book we do not consider second order Lagrangian systems, their appropriate setting is also described, together with a new proof of the equivalence between the Hamiltonian and the Lagrangian index.

Preface

xi

Chapter 4. We start by stating those results of Morse theory that we need for our applications. Then we focus on superquadratic Hamiltonians, proving the existence of periodic orbits which do not necessarily satisfy Rabinowitz condition, using index estimates. We also show how information

on the index can be used to prove the minimality of the period in the convex case. Finally, we show a result about the existence of infinitely many periodic orbits near a non hyperbolic fixed point, in the spirit of the Birkhoff-Lewis theorem.

Chapter S. Here asymptotically linear systems are studied in detail. We start by proving the Morse relations for systems which are non-resonant at infinity, deducing higher dimensional versions of Poincare's last geometric

theorem, then present a statement about subharmonic solutions of twodimensional systems. We also deal with autonomous systems, giving an estimate of the number of their periodic solutions in terms of the Maslov indices of their equilibrium points. Finally, we study systems which are resonant at infinity, under Landesman-Lazer conditions, strong resonance conditions and another kind of assumption.

Chapter 6. We discuss the Arnold conjectures about the number of fixed points of Hamiltonian diffeomorphisms of compact symplectic manifolds. We prove these conjectures for the standard torus and for the complex projective space. We conclude the chapter with a multiplicity result for periodic points on the torus.

Acknowledgments. I wish to express my gratitude to Vieri Benci and to Pietro Majer, from whom I learned most of the mathematics contained in these pages. I am also indebted to Louis Nirenberg and Helmut Hofer for giving me the opportunity to spend a long period of time at the Courant Institute, where I found an ideal environment to write this book. New York


Chapter 1

The Maslov index In this chapter. we will study the properties of an important invariant of linear periodic Hamiltonian systems called Maslov index. Such an invariant

will play an important role in connection with a suitably defined Morse index of periodic solutions of non-linear systems. The Maslov index is, roughly speaking, the number of half windings made by the fundamental solution of a linear Hamiltonian system in the symplectic group. Both non-degenerate systems (systems with no non-trivial periodic solutions) and degenerate ones will be examined. We point out that some authors prefer to call this invariant Conley-Zehnder index, using the term Maslov index for an integer which labels paths of Lagrangian subspaces: with this language, the invariant we study here is a special instance of the Maslov index, adapted to the case of periodic orbits. Our terminology agrees with the one used by Salamon and Zehnder in [SZ92} and by Long (see [Lon97) and the literature cited later on).

1.1

The symplectic group

Definition 1.1.1 Let V be a finite dimensional real vector space. A symplectic form w is a non-degenerate anti-symmetric bilinear form on V. A symplectic vector space is a pair (V, .;).

The basic example is R2, with coordinates (pj, qi .... , pN, qN), endowed with the 2-form dpi A dq;.

wo

1

Chapter 1. The Afaslov index

2

Equivalently, if u v denotes the usual inner product in R2N,

where J:=1 0 0

0

01

The matrix J is skew-symmetric and it satisfies the important identity J2 = -I. We will refer to wo as the standard symplectic form on R2 v. In some of the next chapters, we will also find it useful to order the coordinates in R2N as (PI, , PN, qi, ... , qN), so that J will be written as

J=

%0

lI

0 1 '

where I denotes the N by N identity matrix. Definition 1.1.2 A linear subspace W of a symplectic vector space (V, w) is called symplectic if the restriction of w to W is non-degenerate. Proposition 1.1.1 Let (V, w) be a symplectic vector space. Then V is even dimensional and it has a basis e1, fl,..., eN, fN such that

w(ei, ei) = w(fi, fi) = 0 Vi, j, w(ei, fi) = 0 if i 0 j, w(ei, fi) = 1 Vi. Proof.

(1 . 1)

Choose a vector el 96 0 in V. Since w is non-degenerate, we

can find a vector v E V such that w(el, v) 76 0. We can normalize v to obtain a vector fl such that w(ei, fl) = 1. Since w is anti-symmetric, fi is not a multiple of el; therefore, el and fl span a two-dimensional subspace Vi of V. Notice that Vl is a symplectic subspace of V. If V has dimension two the construction is finished, otherwise we can consider the subspace

W:=(WEVIw(w,u)=0b'uEVI)

.

Also, W is a symplectic subspace of V and it is a complement of Vi : V = o Vi ® W. Therefore, we can apply the same construction to W. A basis el, fl,..., eN, IN which satisfies (1.1) is called a symplectic basis

of (V, w). The standard basis of RZN is symplectic with respect to the standard symplectic form wo.

Definition 1.1.3 Let (VI, wl) and (V2, w2) be symplectic vector spaces. A symplectic isomorphism from (Vl, wl) to (V2, w2) is a bijective linear map cp : VI - V2 such that cp'w2 = wl, meaning that w2(cp(u), cp(v)) = wi (u, v),

bu, v E V1.

In the case (VI, wi) = (V2, w2), cp is called a symplectic automorphism.

I.I. The symplectic group

3

Remark 1.1.1 Proposition 1.1.1 shows that every symplectic vector space of dimension 2N is symplectically isomorphic to (R2N, W.).

The matrices which correspond to symplectic automorphisms of the standard symplectic space (R2N,wo) are called symplectic and they are characterized by the equation

ATJA= J

(1.2)

where AT denotes the adjoint of A. Let E = lei, fi, ... , eN, fN} be a symplectic basis of (6;w). A linear automorphism ;F of V is symplectic if and only if the matrix associated to

p with respect to the basis 6 is symplectic. The set of symplectic automorphisms of (V, w) forms a group, denoted by Sp(V, w). The set of the symplectic matrices is a group, denoted by Sp(2N). Remark 1.1.1 shows that Sp(V w) is isomorphic to Sp(2N). Since symplectic automorphisms preserve w, they also preserve the 2Nform or = wA .. nw, N times the wedge product of w by itself. In the case of the standard symplectic space, a coincides with the standard volume form of R2N, possibly with the opposite sign, and therefore, every symplectic matrix has determinant 1.

Since Sp(2N) is a closed subgroup of GL(2N), it inherits a structure of Lie group (see e.g. [War7l] Theorem 3.42). More directly, the fact that Sp(2N) is a manifold can be proved by noticing that

Sp(2N) = JA E L(2N) 4(A) = J), where L(2N) is the vector space of all real matrices of order 2N and

4(A) := ATJA.

The map 4 is smooth from L(2N) to Skew(2N), the vector space of all skew-symmetric matrices of order 2N. The differential of fi in A E L(2N) is

d4(A)[H] = HT JA + ATJH. If A is invertible. then d4(A) is onto. In fact, if B E Skew(2N), d4(A) [J(A_1)TBT] 2

= B - IBT = B. 2

In particular, d4(A) is onto whenever A E Sp(2N). This proves that Sp(2N) is a smooth submanifold of L(2N).

The tangent space to Sp(2N) at the identity matrix I. i.e. the Lie algebra of Sp(2N), is

sp(2N) := Kerd4'(I) _ {H E L(2N) I HTJ+ JH = 0) .

Chapter I. The Maslov index

4

The matrices in sp(2N) are called infinitesimally symplectic. Rom the theory of Lie subgroups of GL(2N) we deduce that eH

°G

1

j_o

belongs to Sp(2N) if H E sp(2N). Moreover, every symplectic matrix is the exponential of an infinitesimally symplectic one. The first assertion can also be proved directly as follows. The smooth path y(t) = el" is the solution of the Cauchy problem

y'(t) = Hy(t),

1 y(o) =I. If H E sp(2N) then Wt [_t(t)T Jy(t)] = -r(t )T HT J? (t) + y(t)T JH7(t) = o.

Since y(0) is symplectic, y(t) must be symplectic for every t E R.

1.2 1.2.1

The Maslov index in dimension 2 The topology of Sp(2)

In this section, the symplectic group of R2 is studied in detail. On R' the standard symplectic form w is nothing else but the usual area form dpAdq. Therefore, an automorphism is symplectic if and only if it preserves

the oriented areas: the symplectic group Sp(2) coincides with SL(2), the group of real matrices two by two having determinant one. Every invertible matrix A can be decomposed in polar form

A = PO, P:= (AAT) J, 0:= P - L A, where P is symmetric and positive and 0 is orthogonal. If A E Sp(2), P must have determinant one, so P E Sp(2). Thus, also 0 E Sp(2). Hence, 0 belongs to the group of rotations SO(2), the orthogonal matrices with determinant one. The group SO(2) is isomorphic to U = {z E C I IzI = 1). Let U : Sp(2) -+ U be the map which associates to every symplectic matrix the rotation angle of its orthogonal part: U(PO) := eee

-sing ) if O = C cos0 sin8 cosA

Since P and 0 in the polar decomposition are unique and depend continuously on A, Sp(2) is homeomorphic to the topological product of U and

1.2.

The kfaslov index in dimension 2

5

the set of two by two symmetric and positive matrices with determinant one. Call P(2) the latter set. The set P(2) is homeomorphic to the plane because every matrix in P(2) is the exponential of one and only one matrix in the two-dimensional vector space consisting of symmetric infinitesimally symplectic matrices. However, we will build a different homeomorphism, which will make some later calculations easier. Such parameterization is due to Gel'fand and Lidskii [GL58] (see [Lon97] for a different parameterization).

The trace of P is the sum of its eigenvalues, which are positive and whose product is 1. Therefore, tr P > 2 and we can set tr P = 2 cosh r, with r > 0. Then P can be written as

p_

coshr+a

b

b

cosh r - a

There is a relationship between a and b, determined by the determinant condition

1 =detP=cosh 2r-a2-b2. Then b can be expressed in terms of a as

b2 = cosh2 r - a2 -1 = sinh2 r - a2. The above equation makes sense if and only if Iai < sinh rJ. Therefore, we can set a = cos a sinh r, with a E R. Then b2 = sin2 a sinh2 r and we obtain a one-to-one parameterization of P(1) if we set b = sinasinhr and we let a vary in [0, 27r[. If we consider (r = IzI,a = argz) as polar coordinates on the complex plane, a homeomorphism between C and P(2) is given by

P -( cosh r + cos a sinh r sin a sinh r

sin a sinh r

cosh r- cos a sinh r J

In order to draw better pictures, it is more convenient to parameterize P(2)

by the open unit disk D = {z E C I IzI < 1}. To do this, it is enough to consider polar coordinates (r = IzI, a = arg z) on D and to set r = tanh2 r. Therefore, Sp(2) is homeomorphic to R/2arZ x D, the product space of the circle by the open disk, i.e. to the interior of the full torus, as it is shown in Fig. 1.1. It is not hard to check that the above homeomorphism is actually a diffeomorphism.

1.2.2

The rotation function on Sp(2)

We will always consider coordinates (8, r, a) on Sp(2), as in the previous section. Notice that the group of rotations SO(2) corresponds to the circle

{(B, r, a) E R/2nZ x D [ r = 0}.

Chapter 1. The Maslov index

6

Figure 1.1: The parameterization of Sp(2).

Thus SO(2) is a deformation retract of Sp(2). Moreover, U(O, 0, o) = e'9, so U restricts to the standard isomorphism from SO(2) onto U. The function U is not symplectically invariant, meaning that, in general, if A, M E Sp(2),

U(M-SAM) 0U(A). Moreover, U is no longer a homomorphism on Sp(2), and in general, U(A2) jA U(A)2.

In order to avoid this problem, we are going to define a symplectically invariant map p : Sp(2) -+ U,

homotopic to U, which is still not a homomorphism but has the property

that p(A") = p(A)n, for every A E Sp(2). This map will be called the rotation function. The eigenvalues of A E Sp(2) must be of the form A, z, where A E AUU. An eigenvalue A iA t1 must therefore be simple. The eigenvalues 1 and -1 are always double.

Consider the matrix G := -iJ, which is Hermitian with respect to the standard Hermitian product of C2. Denote by A- = AT the transposed conjugate of A. Notice that a real matrix A is symplectic if and

only if A'GA = G. Assume that A E Sp(2) has eigenvalues A # ±1 and A on the unit circle and that f and are the corresponding eigenvectors. Then

(Gf,a) = (AsGAf,Z) = (GAd,A!) = A2(Gf,f)

Since A iA ±1, (G f , a) must be null. So {} is a G-orthogonal basis of C2. Hence (G f , f) and (Ge, Z) are real and not zero. This fact justifies the following definition.

1.2. The Maslov index in dimension 2

7

Definition 1.2.1 If A E U \ { -1,1 } is an eigenvalue of A E Sp(2) and t is the corresponding eigenvector, the Krein sign of A is the sign of (Ge, t).

Since G has signature (1, 1), if the eigenvalue A E U \ {-1,1} is Kreinpositive, the eigenvalue A is Krein-negative. Sometimes it is useful to consider the double eigenvalue A = ±1 as a pair of eigenvalues, one of which

is Krein-positive, the other Krein-negative. We can define the rotation function p : Sp(2) - U as

p(A)

A

if A E U \ {-1,1) is the Krein-positive

1

eigenvalue of A, if the eigenvalues of A are real and positive, if the eigenvalues of A are real and negative.

_

-

-1

The function p is symplectically invariant by construction. In order to see that p is continuous, notice that

p(A) _ A where A is any eigenvalue of A, in the case A E R, and it is the Kreinpositive eigenvalue, in the case A E U \ {-1,1}. The rotation R(6) _

(

cos 8 sin 8

- sin 8 cos 8

has eigemalues e'8, e`9 and

cos8+isine

sin8-icose } is an eigenvector corresponding to e'8. A direct calculation shows that

(Gt, ) = 2, so ei6 is Krein-positive and p(R(8)) = ei9. Therefore, p coincides with the map U on the subgroup SO(2). If A is an eigenvalue of A, Ak is an eigenvalue of Ak, the eigenvectors being the same. Therefore, p(A5) = p(A)k.

However, since the eigenvalues of the product of two matrices need not be the products of the eigenvalues, p is not a homomorphism on the whole Sp(2). In order to study the function p in terms of the coordinates (9, r, a), we must find the eigenvalues of A, so we must solve

det(AI - A) = A2 - (tr A)A + 1 = 0


8

Figure 1.2: The rotation function p. for A. An easy computation shows that the discriminant of this polynomial is

'=(trA)2-4=4cosh2reos20-4. The matrix A has a double eigenvalue ±1 if and only if A = 0, which turns out to be equivalent to r = sine 9,

because cosh2 r = 1/(1 - tanh2 r) = 1/(1- r). So the set of symplectic matrices with a double eigenvalue is the light surface of Fig. 1.2. It consists of two connected components (remember that the full torus has no boundary). One component contains I and thus it consists of matrices with eigenvalue 1. The other component contains -I and consists of matrices with eigenvalue -1. The inequality :1 > 0 is equivalent to r > sin2 9.

Therefore, p = ±1 outside the light surface. By continuity, p = 1 on the right side, and p = -1 on the left side. If A is in the region inside the light surface, A is less than zero and A must have taro eigenvalues A, A E U \ { -1,11. The region inside the light surface consists of two components, one of which contains the rotations of angle 0 with sin 9 > 0, the other the rotations of angle 9 with sin 9 < 0. Call these regions f1+ and fl-, respectively. The map p is continuous, it never takes the values ±1 in the region inside the light surface and p(U) = e`e

whenever U is a 9-rotation: therefore, p must take values on the upper half circle U+ = {z E U I Im z > 01 in fl+, and on the lower half circle

U'= {zEUIImz1

where F,, := E,,, if JAI = I and FA := Ea 0 Ea-) if JAI > 1. By step 2, the above decomposition is g-orthogonal. Therefore, g must be non-degenerate on each of the spaces F,,. 4. Let A be an eigenvalue of A E Sp(2N: C) outside the unit circle, with algebraic multiplicity d. Again by step 2, g(v, v) = 0 for every v E Ea. Then g restricted to the 2d-dimensional space Fa has a d-dimensional isotropic subspace. We conclude that g has signature (d, d) on F,,. 5. On the contrary, g may have any signature on F., = E,, in the case IAI = 1. This fact justifies the following definition.

Definition 1.3.1 Let A be an eigenvalue on the unit circle of a complex symplectic matrix. The Krein signature of A is the signature of the restriction of the Hermitian form g to the generalized eigenspace Ea.

6. Now let A be a real symplectic matrix. Since A is real, it has pairs of eigenvalues A and A, with the same Jordan form on the corresponding generalized eigenspaces. If we put this fact together with what we proved at step 1, we get the following result: if A E Sp(2N) has an eigenvalue A, then it has also the eigenvalues A, a-1 and A-'. Moreover, A has the same Jordan form on each generalized eigenspace. Therefore, the eigenvalues of a real symplectic matrix come out in groups, according to the following list:

1.3. The Alaslov index in dimension 2N

13

(i) the real eigenvalues different from ±1 are always in pairs A. a;

(ii) the eigenvalues on the unit circle. different from ±1, are always in pairs A. A;

(iii) the eigenvalues away from the real line and from the unit circle come

out four by four: A. A. A-t and A': (iv) the eigenvalues 1 and -1 always have even algebraic multiplicity.

The last assertion follows from the fact that det.A = 1. 7. Assume that .4 is real symplectic. If the eigenvalue A E U has Krein signature (p. q) then the eigenvalue A has Krein signature (q. p). In fact, Ex = Ea and

9(P. P) = (-IJi\, i) = (iJr. r) _ -9(v. r) = -9(v. t'). In particular. the Krein signature of the eigenvalue I or -1 has always the form (p, p).

If the real symplectic matrix .4 has an eigenvalue A on the unit circle of Krein signature (p. q), it is often convenient to say that .4 has p + q eigenvalues A, and that p of theist are Krein-positive, while q of them are Krein-negative. With these notations. if 2k is the total algebraic multiplicity of the eigenvalues of A on the unit circle, we can say that there are Al

..... Ak Krein-positive eigenvalues and A, ..... Ak Krein-negative eigen-

values.

Recall from section 1.1 that the space of infinitesimally symplectic matrices sp(21') consists of the matrices H such that

HTJ+JH =0. Moreover, ell is symplectic for every H E sp(2N) and every symplectic matrix can be written in this way. Arguing as before. it is easy to describe the spectrum of an infinitesimally symplectic matrix H. In fact. H belongs to srp(2N) if and only if

J.IHTJ=-H. Therefore. if A is an eigenvalue of H. so are A. -A and -A, all with the same multiplicities. Hence, three situations are possible: (i) the nonzero real or purely imaginary eigenvalues occur in pairs:

{A.-A}.

AERUiR\{0}:


14

(ii) all the other nonzero eigenvalues occur in groups of four:

A E C \ (R u iR); (iii) the eigenvalue zero always has even algebraic multiplicity. To each eigenvalue A of H there corresponds an eigenvalue ell of A = eH.

The reader can easily check that the above description of the spectrum of H agrees with the description of the spectrum of the symplectic matrix A. The Krein signature of the eigenvalues of H is defined as before.

Definition 1.3.2 Let A be a purely imaginary eigenvalue of an infinitesimally symplectic matrix. The Krein signature of A is the signature of the restriction of the Hermitian form g to the generalized eigenspace Ea.

With the convention of repeating the eigenvalues according to their multiplicity, if 2k is the total number of purely imaginary eigenvalues of H, we can say that there are Al, ... , Ak Krein-positive eigenvalues and -A1, ... , -Ak Krein-negative eigenvalues. Krein theory, of which this section was only an introduction, is an important tool for the stability question of periodic solutions (see [Eke90], [Kre5O], and [YS751).

1.3.2

Normal forms of semi-simple symplectic matrices

Let A be a semi-simple symplectic matrix, meaning that the algebraic and geometric multiplicity of its eigenvalues coincide. By the discussion of the previous section, there exists a decomposition of R2N into symplectic subspaces

R2N

= V(1) e V(2) ED 1,(3)

where:

1. VM is the real part of the sum of the eigenspaces of A corresponding to the eigenvalues on U. Therefore. Vil) has the symplectic decomposition

trtl) = E1 n R2N ©E_ 1 n R2N s ®(EA a E.X) n R2N. AEU Im A>O

2. V(2) is the real part of the sum of the eigenspaces of A corresponding

to the real eigenvalues different from ±1. Therefore, V(2) has the symplectic decomposition

02) =

®(EAaEA-,)nR2N.

AER IAA>1

1.3. The Maslov index in dimension 2N

15

3. V(3) is the real part of the sum of the eigenspaces of A corresponding to all the other eigenvalues. Therefore, 03) has the symplectic decomposition

(E), ®Ez ») Ea-, 0 Ez-.) n R IV.

V(3) lal>1

Im A>O

Since A is semi-simple, V(1), V(2) and 033 actually have the symplectic decomposition

®Vai), j=1,2,3. 1-1

where the spaces VV ) are two dimensional for j = 1, 2 and four dimensional

for j = 3. Now we want to examine the form of A on each of the spaces ti w. Call A;j) the restrictions of A to the above subspaces.

1. The space V` 1) corresponds to a pair of eigenvalues ese, a i9 in U, with 8 E R. Then there exists a symplectic basis of t;1} such that cosO

A;1 =

sin8

-sin O cos8

2. The space V; 2) corresponds to a pair of real eigenvalues p, p' 1, with IpI > 1. Then there exists a symplectic basis of V 2) such that A(2) =

(0

11

0

p-1

3. The space I;3} corresponds to a group of eigenvalues

peie, pe-ig.p lei' p 1e-s

,

with 9 E R\ irZ, p E R, (pI > 1. Then there exists a symplectic basis of V,(3) such that

A(,3) =

'

p COs 8

0

-p sin 8

0

0

0

-p-1 sin8

psin8

p-1 cos8 0

pcos8

0

0

p-1 sing

0

µ l cos8

Chapter 1. The llaslov index

16

Definition 1.3.3 We will say that a semi-simple symplectic matrix A is in normal form if where Ai has one of the forms listed above. The above discussion can be reformulated in the following way.

Proposition 1.3.1 Let A be a semi-simple symplectic matrix. Then there exists a symplectic matrix 1I such that .4 = 111 B AL where B is a semisimple symplectic matrix in nonnal form. It is not difficult to derive normal forms for symplectic matrices which are not semi-simple. However, we will be able not to use these normal forms by approximating an arbitrary symplectic matrix with a semi-simple one. Indeed, the following result holds.

Proposition 1.3.2 The set of semi-simple symplectic matrices is dense in Sp(2N). Proof. Let A E C \ (R U U) be an eigenvalue of A E Sp(2N) with algebraic multiplicity m. Choose a basis ti. - - .. C,,, of EA such that

(A! - A)t;1 E span

A6 = A6 ,

tl-1 I.

By step 2 and step 3 of section 1.3.1. Ea and

are y-isotropic subspaces

of Fa = Ea Ex-.. and y is non-degenerate on FA. Therefore, we can find a basis rtt .... , %, of EX_, such that &j.

For e > 0 let Af, be the identity matrix on

err..... rlrn 2 .... m . rjo..... I)m }.

span

and on E,,. for p q {A, X. A111tF.1

while

= (1 +c)41,

AIEZ1 = (1 +E)EL.

1

3IEr_h

1

_

= + 1 +f'll Then Al, E Sp(2 V) and the algebraic multiplicity of the eigenvalue A of 11 U induces an isomorphism of fundamental groups. Moreover, the fundamental group of Sp(2N) is infinitely cyclic: v'1(Sp(2N)) = Z.

1.3.4

The rotation function on Sp(2N)

Our aim now is to define the rotation function on Sp(2N), generalizing the construction we used for Sp(2).

Definition 1.3.4 Let A1i ... , A2N be the eigenvalues of A E Sp(2N), repeated according to their multiplicity. The eigenvolue A, is said of the first kind if either lA;I < I or jA;I = 1 and A, is Krein-positive. Otherwise, it is said of the second kind. Therefore, we can always order the eigenvalues of A as

A1i...,AN,Ai 1 ,...,AN 1,

... , AN are of the first kind. With such an ordering, the rotation function p : Sp(2N) --+ U is defined as where A1,

N P(A)

_

A

fa

An alternative equivalent definition is

p(A) _ (-1)m

H pEa(A)nU\{f1)

priu),

(1.7)


20

where 2m is the total algebraic multiplicity of the real negative eigenvalues of A and (p(p),q(p)) is the Krein signature of the eigenvalue p E U.

Denote by I'(n) the quotient topological space obtained by C", under the equivalence relation given by coordinate permutations: r(n) is the space of unordered n-uples of complex numbers. The continuous dependence of

the eigenvalues of a matrix in L(n) can be stated saying that the map from L(n) to r(n) which assigns to every matrix the unordered n-uple of its eigenvalues, counted according to their multiplicity, is continuous (see [Kat8O] section 11.5.2). An analogous statement holds if we consider only the eigenvalues of the first kind of a symplectic matrix.

Lemma 1.3.3 The map Sp(2N) -+ I'(N) which assigns to every symplectic matrix the unordered N-uple of its eigenvalues of the first kind, is continuous. Proof. Let A be an eigenvalue of the first kind of A, with algebraic multiplicity m. If )A) < 1, every symplectic matrix in a suitably small neighborhood of A has eigenvalues ps, ... , ph, close to A, with total multiplicity m, of the first kind. There remains to consider the case of an eigenvalue A

on the unit circle. Let (p, q) be its Krein-signature, so that p + q = m. If we slightly perturb A, the eigenvalue A will split into eigenvalues pl, ... , ph such that )p;) = 1 , and eigenvalues Vi , vi 1 , ... , Vk, Vk t such that Iv;) < 1. The sum of the algebraic multiplicity of all these eigenvalues is m and, by continuity, g has signature (p, q) on the sum of their generalized eigenspaces.

Let (pi, q,) be the Krein-signature of pj and let dj be the multiplicity of vj. Then the eigenvalues of the first kind bifurcating from A are

p1,...,phiVl,...,Vk, with total multiplicity h

pj

k

+Fd,.

By step 4 in section 1.3.1, g has signature (d J, dj) on C EE, -1. hence the above number equals p, proving that the total multiplicity of the eigenvalues 0 of the first kind bifurcating from A is p, as we wished to prove.

Given an interval I C R and a continuous path y : I - Sp(2N), it will be often useful to build continuous functions Al, ... , AN : I -+ C such that Al (t), ... , AN (t) are the eiegenvalues of the first kind of y(t), repeated according to their multiplicity. Such functions exist because of the above lemma and of the following result, whose proof can be found in Kato's book [Kat8O], Theorem 11.5.2.

1.3. The Maslov index in dimension 2N

21

Proposition 1.3.4 Let A : I -- r(N) be a continuous path. Then there exist continuous functions Al, ... , AN : I -+ C such that A(t) = [(AI (t), ... , AN(t))]-

Let (V, w) be a real symplectic space and let t: = (ei, fl,. ... eN, IN) be a symplectic basis of (V w). If V is a symplectic automorphism of (V, W), let A E Sp(2N) be the matrix which corresponds to rp with respect to the basis C. Then we can extend the rotation function as a map

p:Sp(V,w)-+U, by setting p(rA) := p(A).

In order to see that such extension is well-defined, we must check that its value does not depend on the choice of the symplectic basis E. If E' is another symplectic basis of (V, w), then the matrix associated to op with respect to £' has the form

A'=M.AM-', for some M E Sp(2N). Let A be an eigenvalue of A, with generalized eigenspace EA. Then ME), is the generalized eigenspace of A' corresponding to the eigenvalue A. Since M is symplectic, g has the same signature on Ex

and on MEa. Therefore, p(A) = p(A').

Theorem 1.3.5 The rotation function p satisfies the following properties.

(i) (Continuity) The function p is continuous. (ii) (Symplectic invariance) If tG isomorphism then

:

(4i,w1) -+ (V2rw2) is a symplectie

for every ;p E Sp(Vi,wi). (iii) (Splitting) If (V. w) = (VI () V2, WI

w2), then

p(ip) = p(im)p(), f o r + p E Sp(V, w) o f the f o r m cp(zi

( - .3

).

(iv) (Value on the unitary group) If A E Sp(2N) C O(2A') and U is the corresponding matrix in U(N), then p(A) = det U.


22

(v) (Normalization) If 9 E Sp(T w) has no eigenvalue on the unit circle then

AP) = ±1. The above properties characterize the family of functions p. Moreover: (vi) (Homotopy) The map p induces an isomorphism of fundamental groups

p.: rri(Sp(V,w)) -+ rri(U)(vii) (Iteration) If cc E Sp(V, w) and k E Z, then

p(54k) = A'A'. (i). It is enough to prove the continuity of p on the standard Proof. symplectic group Sp(2N), which follows immediately from Lemma 1.3.3. (ii). This follows from the same argument used to prove that the definition of p(V) does not depend on the choice of the symplectic basis. (iii). Assertion (iii) follows immediately from the definition of p. (iv). If A is symplectic and orthogonal, all its eigenvalues have modulus one and

p(A) = fl

AP{-'),

AEo(A)

where (p(A),q(A)) is the Krein signature of A. Let F} and F- be the elgenspaces corresponding to the eigenvalues i and -i of J, respectively. Recall that the restriction of A to F+ is the unitary operator associated to A. We denote such restriction by U. Since AJ = JA, the generalized eigenspaces EA, A E a(A), are J-invariant and

F+= ® EAnF+. AEo(A)

The Hermitian form g =

is positive on F+ and negative on F.

so p(A) = dim EA n F+. Hence we get

det U = 11 Adim 6anF* = AEo(A)

[I

Ap(ai.

AEo(A)

proving the claim.

(v). Assertion (v) follows immediately from identity (1.7).

1.3.

The Afaslov index in dimension 2N

23

To see that the properties (i). (ii). (iii), (iv) and (v) characterize the family of functions p, let 13 be another family of functions satisfying the same conditions. We will show that 13 = p. Since the set of semi-simple symplectic automorphisms of (T. w) is dense

in Sp(V, w), by Proposition 1.3.2, the continuity of p and 13 allows us to verify the equality 13 = p only on such automorphisms. By the symplectic invariance and the splitting properties, it is enough to check the value of 13 on the three symplectic matrices in normal form

AI)

(

sing

cos9 }

I

A(2)=p 0 0

)A

%'lµ.e =

p E R,

JA

0 p-1 cos 9

-psine

0

0

0

-µ 1 sin g

psin9

0

pcos9

0

0

p-1 sing

0

pcos9

OE R,

.

O,, E R.

p`1 cose

Property (iv) guarantees that de =

The set of matrices of the second kind consists of two connected components, one for it > 0 and the other for p < 0. By property (v), 13 must take the value 1 or -1 on these matrices. Since 13 is continuous and 13(12) = _(Ao1)) = 1,

13(-12) = 13(A(')

we deduce that p(A(2)),

if p > 0,

P(=1N2)) = -1 = p(A(2)),

if p < 0.

13(A(,2)) = 1

Finally, the set of matrices of the third kind is connected and it contain the identity matrix. Again, by assumption (v) and by the continuity of 13, 13(A ;,3g) = 1 = p(A(3e)

(vi). It is enough to verify assertion (vi) on the standard synplectic

group. Identifying U(N) with Sp(2N) f1 O(2N), we have the inclusion map

U(N) y Sp(2N).

Chapter 1. The :lfaslov index

24

Since Sp(2N)

U (IV) x P(2N) and P(2N) is simply connected,

i.: 7rl (U(N)) -+ irl(Sp(2N))

is an isomorphism. The determinant function det : U(N) - U induces an isomorphism of the fundamental groups. Since p o i = (let, also p. is an isomorphism.

(vii). Assertion (vii) easily follows from the fact that Ak is an eigenvalue of qk if A is an eigenvalue of p.

The rotation function was introduced in a pioneering paper by Gel'fand and Lidskii [GL58]. Theorem 1.3.5 is due to Salanion and Zehnder [SZ92]. Actually, it is possible to avoid the introduction of the rotation function. replacing it by the determinant of the unitary part in the polar decomposition (this is the approach followed by Long and collaborators. see for example [Lon90], [Lon97]). (LZ90]. The advantage of the rotation function is that it is a symplectic invariant and it satisfies p(A") = p(A)". These properties turn out to be useful for establishing certain facts about the Maslov index.

1.3.5

Decomposition of Sp(2N)

As in the two-dimensional case. it is useful to divide Sp(2N) into three subsets:

Sp(2N)+

{ 4 E Sp(22V) I det(I - .4) > 0) .

Sp(2N)-={AESp(21V)det(I-.4) 2 with 9i (A) = 0 equals the number of the indices j > 2 with 9i (A) = 2ir. Since (I-)) vanishes on Z, the real eigenvalues give no contribution; all the other eigenvalues of the first kind which do not have modulus one come out in pairs (A, A), so they cancel each other and only the eigenvalues of modulus one are relevant in identity (1.10).

0

1.3. The -Alaslov index in dimension 2N

1.3.7

31

The Maslov index of degenerate paths in Sp(2N)

It is useful to define the Maslov index also for paths y such that y(1) E Sp(2N)°. While the definition of the Maslov index for non-degenerate paths is natural, because it labels the connected components of S' (2N), the extension to degenerate paths is somehow arbitrary. Since we would like the Maslov index to coincide with the Morse index of some suitable self-adjoint operators, and since the Morse index is lower-semi-continuous, we choose

to extend the function p on S(2N) so that it is a lower-semi-continuous function. We remark that for other purposes. different extensions look more appropriate (see [RS93]).

Notice that the quantity A, (y) depends continuously on -F E 8(2N),

while the function R : Sp(2N)' -- R has no continuous extension on Sp(2N). Let R : Sp(2N) -a R be the maximal lower-semi-continuous function such that R < R:

R(A) =

liBminf BESp(2N)'

R(B),

Clearly R = R on Sp(2N)'. By (1.9). IRI < N on Sp(2N)', so

IR(A)I < N. VA E Sp(2N).

(1.11)

Definition 1.3.7 Let I E S(2N). The nullity of 7 is the non-negative integer

v(j) := dim Ker (I -'y(1)). The Maslov index of 7 is the relative integer

Since S'(2N) is dense in S(2N), we have the equivalent definition

I (7) = lim inf p(13),

(1.12)

3ES'(2N)

which shows that p(y) is an integer. The function p is the maximal lowersemi-continuous function on S(2N) which coincides with the Maslov index on S* (2N).

Definition 1.3.7 is due to Long: it was first given in a different, but equivalent, way in [Lon90], and in a form similar to the above one in (Lon97].

Theorem 1.3.11 The quantities R, Ai and p are symplectic invariants. More precisely, if M E Sp(2N),

R(M-'AM) = R(A), VA E Sp(2N), Aj(M-'yM) = t 1(M), p(M-'yM) = p(M). V 7 E S(2N)-

Chapter 1. The Nlaslov index

32

The assertion for O1 follows from the symplectic invariance Proof. of the rotation function. We shall prove that

1Z(M-'AM) = R(A), V.4 E Sp(2N)'.

(1.13)

Let rya be a path connecting A to either -I or W in Sp(2N)'. Then M-1YAM connects M`AM to either -I or M`TVM in Sp(2N)'. In the second case, we can join M'1WM to 4V by joining M to I in Sp(2N). Notice that p e p(W) along this path. Then (1.13) follows from the symplectic invariance of p:

P(M-lyA(t)M) = P(yA(t))It is easy to check that the lower semi-continuous extension of a symplectically invariant function remains symplectically invariant. Hence R is syzn0 plectically invariant, and so must be p.

1.4

The Maslov index of a linear Hamiltonian system

A linear one-periodic Hamiltonian system in R2N has the form

z'(t) = JB(t)z(t)

(1.14)

where B(t) is a one-periodic path of symmetric matrices. Clearly, systems with arbitrary periods can be thought of as having period one by rescaling the time variable. Hence, we will deal only with one-periodic systems. Let y(t) be the fundamental solution of (1.14), i.e. the solution of the matrix differential problem

J y'(t) = JB(t)z(t), l y(o) = I. Then y(t) is symplectic for every t. This follows from the fact that JB(t) is infinitesimally symplectic. Otherwise, it can be verified directly noticing

that Wt (y(t)T Jy(t)J = y(t)T B(t)T JT Jy(t) + y(t)T JJB(t)y(t) = 0.

and that y(0) is symplectic. Definition 1.4.1 The Floquet multipliers of the one-periodic system (1.14) are the eigenvalues of y(1).

1.4. The 'llaslov index of a linear Hamiltonian ss stem

33

Definition 1.4.2 The periodic linear Hamiltonian system (1.14) is degenerate if 1 is a Floquet multiplier. In the opposite case. it is non-degenerate.

Therefore, the fundamental solution of a non-degenerate system is a non-degenerate path in the symplectic group. Degenerate systems are also called resonant systems.

Definition 1.4.3 The nullity of the linear periodic Hamiltonian system (1.14) is the nullity of its fundamental solution. It is denoted by

v(B) := The Maslov index of system (1.14) is the Maslov index of its fundamental solution. It is denoted by

ll(B) := µ(i). Therefore, the nullity of a linear periodic Hamiltonian system is the number of its linearly independent one-periodic solutions. Since the linear periodic Hamiltonian system (1.14) is determined by a loop of symmetric matrices B, we can endow the space of linear periodic Hamiltonian systems with the compact-open topology on the space of loops of symmetric matrices. The space of non-degenerate systems is open. Theorem 1.3.9 can be restated in the following way.

Corollary 1.4.1 The Maslov index is continuous on the space of nondegenerate linear periodic Hamiltonian systems. Two systems lie in the same connected component of the space of non-degenerate linear periodic Hamiltonian systems if and only if they have the same Maslov index.

1.4.1

Iteration formulas

If a system is one-periodic, it is also n-periodic for every n E N. Therefore. we can consider its nullity and its Maslov index at time n.

Definition 1.4.4 Let 7 be the fundamental solution of system (1.14) and set yn(t) = ,(nt). The nullity at time n of system (1.14) is the nullity of I-,,.

It is denoted by

P. (B) = v(7.). The Maslov index at time n of system (1.14) is the Maslov index of y,,. It is denoted by An(B) := µ( 1'n) The Floquet multipliers at time n are the eigenvalues of y(n) = y(1).


34

The above definition makes sense also for negative integers n. By the uniqueness property of Cauchy problems, the fundamental solution y of a one-periodic linear system satisfies

y(n + t) = y(t) }(1)".

Vn E Z, Vt E R.

(1.15)

In particular, y(n) = }(1)", so the Floquet multipliers at time n are just the n-th powers of the Floquet multipliers at time 1. The nullity at time n is always larger than the nullity at time 1, and it is strictly larger when there is some Floquet multiplier which is a nontrivial root of 1. Formula (1.15) reflects into a nice behavior of the rotation function

over the fundamental solution. As before, let d : R i R be a continuous function such that P(7(t)) = e15

,

and let 6(0)

6(t) 7r

Then the Maslov index at time n is

u" (B) = A,, (,y) + R(7(n)) _ ,n(y) +R(y(1)").

(1.16)

The following formula holds.

Proposition 1.4.2 A,, (-y) = n-I (y) for every integer n. Proof.

Consider the homotopy 7A(t)

y((1 - A)t),

t E [0,1), A E [0,1],

so that yo = 71(0.11 and yi =- I. We can extend the paths ya over all R by setting -j,\(t) = 7X(t - n)ya(1)" for t E [n,n + if. Then y.

is continuous on (A, t) E [0,1) x R. By construction, yo = 7,

yt = I and yx(n + t) = y,,(t)ya(1)", Yn E Z, tilt E R.

(1.17)

Let da : R i-+ R be the unique continuous function such that P(la (t)) = ei6a «1

and

6,\(O)=O.

By (1.17) and by the iteration property of the rotation function (Theorem 1.3.5 (vii)), p(7a(n)) = p(ya(1))",

1.4. The Maslov index of a linear Hamiltonian system

35

so (1.18)

6,% (n) - nbx(1)

must be an integer multiple of 2a. Moreover, the quantity (1.18) depends continuously on A, hence

An(7) - nA,(7) = ![bo(n) - ndo(1)] = -[b1(n) - n6i(1)], which vanishes because 61 = 0.

O

Formula (1.16), together with the above proposition and inequality (1.11), implies that the Maslov index at time n grows more or less linearly with n. The rate of this growth is called mean winding number. Definition 1.4.5 The mean winding number of the linear Hamiltonian System (1.14) with fundamental solution -y is the real number T(B) := lim n

µn(B) _ n,+ n

An(7) n - Ol(7)

Theorem 1.4.3 Let B be a one-periodic loop of symmetric matrices. Then

l p.(B) - nr(B)l < N,

(1.19)

the inequality being strict for every n such that the system has at least one Floquet multiplier at time n different from 1.

Proof

Let -y : R - Sp(2N) be the solution of

J 7' = JB(t)7, 1

7(0) = I.

By Proposition 1.4.2,

p. (B) - nr(B) = A.(-t) + R(7(n)) - n01(7) = R(7(n)) So (1.19) follows form (1.11) and the strict inequality comes from the fact that 1 Rl < N on symplectic matrices with at least an eigenvalue different from 1. Indeed, if A E Sp(2N) has an eigenvalue different from 1, we can find a positive number e < 7r/2 and a matrix A, E Sp(2N)' such that

R(A)l
2: for this reason, in the paper by Conley and Zehnder the case N = 1 is excluded. Then Long and Zehnder [LZ90] introduced the approach we also have followed, which is essentially analogous to the one used by Gel'fand and Lid-

ski! (this old paper was probably little known at that time). The extension to degenerate paths is due to Long [Lon9O].

42


An extensive study of the Maslov index and its iteration properties has been carried on by Long and his collaborators (see [DL97], [HL00], [LD00], [LL97], [Lon9l], [Lon97], [Lon99a], [Lon99b], [Lon00b]). The fine properties discovered have provided beautiful applications to Hamiltonian systems, some of which will be the subject of the following chapters.

Chapter 2

The relative Morse index As we will see in Chapter 3, the action functional given by a Hamiltonian system has the form (2.1) Ax) = 2 (Sox, x) + b(x).

where a varies in a real Hilbert space E, with inner product So is an invertible self-adjoint linear operator on E and the nonlinearity b has compact gradient (it maps bounded sets into relatively compact sets). Both the positive and the negative eigenspaces of So turn out to be infinite dimensional, so critical points of functional f have infinite Morse index and co-index. However, the special form (2.1) allows us to define a relative Morse index, which will share many of the properties of the usual index. Relevant concepts here are the notion of commensurable subspaces and of Fredholm pairs. We will also compare our definition of the relative Morse index with another one, involving finite dimensional reductions. Functionals of the above form appear quite often in variational calculus, so it seems useful to develop these concept in full generality, postponing until Chapter 3 the applications to Hamiltonian systems.

2.1

Commensurable spaces and relative dimension

Let E be a real Hilbert space, with inner product (-, ) and related norm fi. Q.

If V is a closed subspace of E, PI., will denote the orthogonal projection

onto V and V1 the orthogonal complement of V. The set of bounded linear operators from the normed space El to the normed space E2 will be denoted by £(E1, E2). The space £(E, E) will be denoted simply by G(E). The image of an operator T E £(E1, E2) will be denoted by R(T). 43

Chapter 2.

44

The relative Morse index

The following notions were introduced in [Abb97] and further developed in [AbbOO], [AMOOb], and [AMOOc].

Definition 2.1.1 Two closed subspaces V and W of E are commensurable if the operator Pv - Pw is compact.

Commensurability is readily seen to be an equivalence relation. The identities

Pv - Pw = PvPW' + PvPw - Pw = (Pwl Pv)' - Pvl Pw,

Pv1Pw=(1-Pv)Pw=(Pw+Pwl -Pv)Pw=(Pw-Pv)Pw, show that V and W are commensurable if and only if both the operators Pwi. Pv and Pv.L Pw are compact.

Definition 2.1.2 Assume that V and W are closed commensurable subspaces of E. The relative dimension of W with respect to V is the integer

dim(W,V):=dim WnV1-dim WlnV. The spaces W n V -L and WI n V above are finite dimensional because they are the spaces of fixed points of the compact operators Pv.L Pw and Pw.L Pv, respectively. Notice that dim(W, V) = dim W - dim V when both V and W are finite dimensional. An example will clarify the meaning of the relative dimension: assume that W := W' ® W,., where W' is a closed s-codimensional subspace of V and Wr is an r-dimensional subspace of VI. Let V, be the (s-dimensional) orthogonal complement of W' in V. Then the operator

Pv -Pw =Pv, +Pw. -Pw. -Pw. =Pv. -Pw. has finite rank, hence it is compact. Therefore, V and W are commensurable and

dim(w,v)=dim wnvl-dim Wlnv=dim W,.-dim V,=r-s. Actually, a general procedure to build an arbitrary subspace W commensurable with V is the following: first build W' := W' ?W,. as above. Then set W := TW', where T is an invertible compact perturbation of the identity (see Proposition 2.1.2).

Remark 2.1.1 The above definitions make use of the Hilbert structure, but commensurability could be defined also in Banach spaces. Two closed subspaces V and W of a Banach space F are commensurable if the quotient

projections V -+ F/W and W - F/V are compact. Assume that F =

2.1. Commensurable spaces and relative dimension

45

V -& t'' is a splitting of the Banach space F into two closed subspaces. If 14' is commensurable with 1', its relative dimension could be defined as

dim It" n t" - codim (It' + t'').

(2.2)

The reader may check that when F has a Hilbert inner product these definitions coincide with those given above. The above remark implies that the notion of commensurability. although given by means of the inner product. is actually independent on the equiv-

alent inner product chosen on E. More precisely, if E is endowed with another inner product whose related norm is equivalent to II . II two subspaces are commensurable with respect to the new inner product if and only if they are commensurable with respect to the original one. We will give another proof of this fact in the next section, where we will also show that the relative dimension does not depend on the equivalent inner product chosen on E. If I. It '. Z are closed commensurable subspaces of E. the following facts will be often used. (i) dim(t. It') _ - dim(IV'. I"): (ii) dirn(Z, V) = ditn(Z. IV) + dim(II", t'):

(iii) t"- and It'- are commensurable and dim (W-L. t"-)

dint(W, U);

(iv) the subspace V+ It' is closed:

(v) Pit" =

E C(tt'. V) is a Fredholm operator of index

ind Iii = dim Ker Pvl` - codim

dim(It , V).

Statements (i) and (iii) follow immediately from the definitions. Since I + (Pty - At-). the operator Pti- + is a compact perturbation of I. so it has a closed image. If r E Tier (Pty + P«-.'. ). we get IIP\ r112 =

r) =

= -IIPtir. r112.

So r E t"1 n It' and Ker (Pty + Pt;-.) = 1"1 n I1'. Since Ps + self-adjoint and has a closed image.

V+ I1"1 J R(A- +

(Ker (Pty +

))1

= (1" (1 I ')1= t' + % so that V + U"- is closed, proving (iv).

.

is

Chapter 2. The relative Morse index

46

The operator Py coincides with the restriction to W of I + (Pv - Pw). Since Pv - Pw is compact, Pµ' has closed image. Moreover,

KerPu' = W n V1,

R(PwY)1 = V n W 1,

so Pv' is a Ftedholm operator of index dim(W, V), proving (v). To prove (ii), notice that

p1 - Pv Pi = PI- Iz - Pv I wPw I z = PvPw.. I zTherefore, Pv is a compact perturbation of the Ftedholm operator Pti'' PK E C(Z,V). Since the Fredholm index is invariant by compact perturbations and it is additive with respect to composition, (v) implies (ii).

2.1.1

Further properties

Let S(E) be the set of all closed subspaces of E. The distance

dist(V.W):=IIPr-PwII makes S(E) a complete metric space, isometric to the subspace of C(E) consisting of the orthogonal projections. If V E S(E), let Sv(E) be the set of all closed subspaces of E which are commensurable with V.

Proposition 2.1.1 Sv(E) is a closed subset of S(E) and the function dim(.. V) : Sv(E) -+ Z is continuous.

Let (14',) C

be a sequence which converges to some must be compact, being the limit of compact operators. This shows that Sv (E) is closed. If W, Z are closed subspaces of E and z E W n Zl, IIxII = 1, then Proof.

W E S(E). Then Pw - Pv = lim,,...,,(Pw -

dist (W, Z) > II

PZSU = IIzII = I.

So, if dist (W, Z) < 1, we get that W n Zl = tV n Z = 1O). Hence. dim(W,, V) - dim(ti , V) = dim IV,, n WI - dim W n W = 0 when dist (W,,, W) < 1. This shows that dim(., V) is continuous on S%- (E).

0 An important issue is to understand what happens to the notion of commensurability and to the relative dimension when a bounded linear operator is applied. Next proposition implies, in particular, that a linear operator of the form identity + compact maps any closed subspace onto a

2.1. Commensurable spaces and relative dimension

47

subspace in the same commensurability class. Moreover, the usual finite dimensional formula

dim R(T) + dim KerT = dim El .

for T E G(E1 . Ej),

has its relative generalization.

Proposition 2.1.2 Let El, E2 be Hilbert spaces, T,T' E C(E1,E2), K := T - T' compact, R(T) and R(T') closed. Then KerT' is commensurable to KerT, R(T) is commensurable to R(T') and dim(R(T'), R(T)) = - dim(Ker T', Ker T).

Since R(T) is closed, T restricts as an invertible operator

Proof.

from (KerT)' to R(T). Denote by S E C(R(T),(KerT)') its inverse; then ST = P(Ker T)1 . The identities P(Ker 7')' PKer T' = ST PKerT'

= ST'PKcrT' + SKPKerT' = SKPKrT' imply that P(KerT)I PKerT' is compact. For the same reason P(Ker T') . PKer T is compact, so KerT' is commensurable to Ker T. The identities PR(T')iPR(T) = PR(T.)..TSPR(T) = PR(T,)i.T'SPR(T) + PR(T')J. KSPR(T) = PR(r),L KSPR(T)

show that PR(T') PR(T) is compact. Similarly PR(T) i PR(T') is also compact, so R(T') is commensurable to R(T).

Set V := (KerT)' and 11' := (Ker T')-. Then V and 11' are commensurable and dim(IV, V)

dim(Ker T', KerT).

Consider TP y and PR(T )T' as operators from W to R(T). Since Tlv V-+ R(T) and T'Iw : 11' -- R(T') are invertible, the above operators are Fredholm and

indTP '' = ind Py', ind PR(T )T' = ind PR (T) ) The identities

TPy -

PR(T))T'

= T[IEt + (Pv - Pw)) - [IE, + (PR(T) - PR(T')))T' = T(P.. - Pw) + K - (PR(T) - PR(T'))T'


48

show that their difference is compact, so they have the same index. Hence. ind PW = ind PR T) ' and

dim(Ker T', Ker T) = - dim(W, V) = -ind PP _ -ind PRINT

dim(R(T'), R(T),

which concludes the proof.

An easy consequence of the above result is the following.

Proposition 2.1.3 Let V, W be closed commensurable subspaces of El and T E C(E1, E2) be an invertible operator. Then TV and TW are commensurable subspaces and dim(TIV, TV) = dim(WW, V). More generally, the same conclusion holds for T E C(E1, E2) with TJv, Taw injective with closed ranges.

Proof.

Consider the operators TPt; and

with closed ranges

R(TPv) = TV and R(TPty) = TW. Since K:= TPt; - TPty = T(Pv Pw) is compact, the thesis follows immediately from Proposition 2.1.2, observing that Ker (TPv) = V' and Ker (TPw) = W1 and using the rule (iii).

The above statement can be used to prove that the notion of commensurability and the relative dimension do not depend on the equivalent inner product chosen on E. We recall that an equivalent inner product on E is an inner product for which there exists c > 1 such that IIxII2 < (x.z) < cI(x112.

`dz E E.

Proposition 2.1.4 The notion of commensurability and the relative dimension do not depend on the equivalent inner product chosen on E.

be an equivalent inner product on E. Then there Proof. Let exists a positive invertible operator T such that (x, y) = (Tx, Y) -

complement of V. Denote by Qv. the Let VI denote the It is orthogonal projection onto V with respect to the inner product easy to see that Qv = T- I PT j vT 1 . Indeed, with such a choice, Q2V

= Qv,

R(Qv) = V.

Qw = TQvT-',

2.2. Fredholm pairs of subspaces

49

the latter identity implying that Qv is self-adjoint with respect to the inner by Proposition 2.1.3 product If V, It' are commensurable in (E, Tit. and Ti 11' are commensurable in (E. also so the operator

Q- -

T(PTjt.

is compact. Hence 17.It' are commensurable in (E.

')).Moreover. since

dim(TF 1'. T1I ) = dimn(1 H'),

by the identities l'° = T-t1'1 and (T-1')1 = T-11'. we get dim X' n 1i" - dim l

n 11' = dim 1' n T-t 1l'1 - dim T_t I''i n U'

=dimTil'nT-ii1'i -dimT-1I'1n7'lIF -ditn(T12l')1 n T^11' = dim(T21-,Till')= diin(l .11'). proving that the relative dimension of l' with respect to IV does not depend on the equivalent inner product.

2.2

Fredholm pairs of subspaces

It is interesting to compare the concepts developed so far with the notion of Fredholm pairs of subspaces, in the sense of Kato. see [Kat8O] section IV.A.

Definition 2.2.1 .4 pair (17. Z) of closed subspaces of E is a Fredholm pair if V+ Z is closed and the numbers dim t' n z, eodim (I' + Z) are finite. In this case, the Fredholm index of (1'. Z) is the relative integer

ind (17, Z) := dint l' n Z - codim (l' + Z). The definition of a Fredholm pair does not involve the inner product. so such a notion can be given also on Banach spaces and it is independent on the equivalent inner product chosen on E. Moreover, the following properties hold, for (l'. Z) a Fredholm pair:

(i) if Y is finite dimensional and t' n 1' = {O). then (t' :- } . Z) is a Fredholm pair of index ind (17. Z) + dim 1;

(ii) (I". Z') is a Fredholm pair of index -ind (1. Z).


50

To prove (i), notice that V ® Y + Z = (V + Z) + I' is closed because I is finite dimensional. It is enough to prove formula

ind (V(D Y,Z)=ind(V,Z)+dim Y

(2.3)

when dim Y = 1, the general case following by an induction argument. Two situations are possible: Y C V + Z or Y n (V + Z) = {O}. In the first case, Y = R(vo + zo), for some vo E V, zo E Z \ V. Then V(9 Y n Z = (V n Z) ®Rzo and V 9 Y + Z = V + Z, so that (2.3) holds. In the second case, (V (B I') n Z = V n Z and codim (V (D Y + Z) = codim (V + Z) - 1, so that (2.3) still holds. Statement (ii) follows from the fact that, if V + Z is closed, then also V-1 + Z1 is closed and

Vl+Z1=(VnZ)1. If V, W are commensurable, then (V, W1) is a FYedholm pair and

ind(V,W1) = dim(V,W).

(2.4)

We remark that, if (VI Z) is Fredholm pair, V and Z1 need not be commensurable. The reader is invited to exhibit a counterexample. The next proposition generalizes both (i) and (2.4). Proposition 2.2.1 Assume that V, W are commensurable and that (17, Z) is a Fredholm pair. Then (W, Z) is a Fredholm pair and ind (W, Z) = ind (V, Z) + dim(W, V). Proof.

Let Z' be the orthogonal complement of V n Z in Z, Z'

VI nZ,and set Z:=Z'®(V1nZl). Then

Vn2={0}, V+2=E, so that E = V ® Z. Let

be an equivalent inner product on E such that V and Z are mutually orthogonal. Denote by V4 the .)-orthogonal complement of V and by Qv the .)-orthogonal projection onto V. Since the notions of commensurable subspaces and the relative dimension do not depend on the equivalent inner product (Proposition 2.1.4), Qv - Qw is compact and

dim( ,V) = dim W nV° -dimW°nV. The operator Q w + Q v- = 1 + (Qw - Qv) is a compact perturbation of the identity, so it has a closed image. Its kernel is Won V. Since Qw is )-self-adjoint, W + V4 D R(Qw + Qv') = [Ker (Qw + Qw' )]° = (W 'Q n V)° = W -+V4.

2.3. Relative Morse index of quadratic forms

51

Therefore, W + Z = W + V° is closed. Moreover,

ind(W,2) =dimWnZ-codim(W+Z) =dimWnV° - dim W°nV=dim(W,V).

(2.5)

Since V n Z and Vi n Zl are finite dimensional, also W + Z is closed. By (i),

ind (14", Z) = ind (W, Z') + dim v n z

=ind(W,2)-dim V1nWl+dim VnZ=ind(W,2)+ind([Z), and the conclusion follows from (2.5).

2.3

Relative Morse index of quadratic forms

Let or : E x E - R be a continuous symmetric bilinear form. We will use the notation a[x]2 = a[z,x] for the related quadratic form. We recall that, by the polarization formula (a[z + y]2 - a[x]2 - a[y]2),

a[-T, y] = 2

every continuous quadratic form uniquely determines a continuous symm ric bilinear form. Therefore, we will make no distinction between symmetric bilinear forms and quadratic forms.

The form a is non-degenerate on a subspace V if for every x E V the identity a[x, y] = 0 for every y E V implies that x = 0. The form or is positive on a subspace V if a[x]2 > 0 for every x E V. It is strictly positive on t' if a[x]2 > 0 for every x E V\ {0}. It is coercive on V if a[x]2 > c{[z[[2, for every z E V, for some positive constant c. Let S be the self-adjoint bounded operator corresponding to a. which is defined by the identity a[x. Y] = (Sx. Y).

We will refer to S as the self-adjoint realization of a on E.

Definition 2.3.1 The symmetric bilinear form a is a Fhedholm form if its self-adjoint realization S is a bounded Fl edholm operator.

Equivalently, we are requiring that zero is not an accumulation point for the spectrum of the bounded operator S and that Ker S is finite dimensional. The notion of Fredholm form is independent on the inner


52

product chosen on E. Indeed, if < , > is an equivalent inner product, there exists a self-adjoint positive invertible operator T E C(E) such

that (x, y) =< Tx,y >. Hence, a[x, y] = (Sx, y) =< TSx. y > and the self-adjoint realization of a with respect to the new inner product is TS. which is Fredholm because T is invertible. Non-degenerate Fredholm forms are called Krein products in [Bog74], while arbitrary Fredholm forms are called weak Krein products in [Bro90]. If o is a Fredholm form, the corresponding self-adjoint operator S determines a unique S-invariant orthogonal splitting E = V+ (S) E. V-(S)

KerS.

such that a is coercive on V+ (S) and -a is coercive on I(S). The above splitting is also a-orthogonal. Operator calculus allows us to define 1'+(S)

and t'-(S) as 1'+(S) = R(XR+(S)).

t'-(S) = R(xa-(S)).

where xR+, respectively XR-, is the characteristic function of ]0. +oc[. respectively I - oc. 0[. With a slight abuse of notation. we will refer to 1 "+(S)

and to V-(S) as the positive and the negative eigenspaces of the operator S or of the quadratic form o. Clearly. the positive and the negative eigenspaces of a quadratic form depend on the inner product chosen on E. Recall that S(E) is the space of closed subspaces of E, endowed with the metric dist (17, W) = IIJ't - PA-II. The positive and negative eigenspaces of an invertible operator S depend continuously on S, as the following result shows.

Proposition 2.3.1 The maps S,-+ V'(S) E S(E) are continuous on the space of invertible self-adjoint operators. Proof.

dist

Indeed

IIXR=(S,) - tR- (S)II

tends to zero if S. tends to S in the operator norm and S is invert ible. An important question is how the positive and negative cigenspaces of a Fredhoim self-adjoint bounded operator S change when S is perturbed by adding a compact operator. The answer is provided by the following result.

Proposition 2.3.2 Assume that ST E C(E) are Fredholm self-adjoint operators such that S - T is compact. Then the positive (respectively the negative) eigenspaces of S and T are commensurable.


53

Proof. The operator p(S) - p(T) is compact for any polynomial p, because it belongs to the two-sided ideal spanned by S - T. By density h(S) - h(T) is compact for any continuous function on E, the union of the spectrum of S and the spectrum of T. Since zero is not an accumulation

point for E, t:R+ and XR- are continuous on E. Hence PV+(T) = XR+(S) - XR+(T) and PV-(s) - Pti'-(T) = XR-(S) - XR-(T) are compact operators.

We are ready to define the relative Morse index of a Fredholm form whose negative eigenspace is commensurable with some given closed sub-

space E-.

Definition 2.3.2 Let

a FVrAholm form. The nullity of a

is the non-negative integer

n(a) = n(S) := dim Ker S.

Assume that the negative eigenspace of S is commensurable with some closed subspace E-. The relative Morse index of a with respect to Eis the integer iE- (a) = iE- (S) := dim(V-(S), E-). The relative large Morse index of a with respect to E- is the integer

iE- (a) = iE- (S) := dim(V (S) a Ker S, E-) = iE- (a) + n(a). When E- = {0}, the relative Morse index and relative large Morse index are just the usual Morse index and large Morse index, and they will

be denoted simply by i(a) = i(S). i'(a) = i'(S). The usual way to apply the above notion will be to fix a non-degenerate Fredholm form (So., ). The invertible operator So determines an

orthogonal splitting E = E+ ® E-, where Et := V*(So). Every quadratic form a whose self-adjoint realization differs from So by a compact operator will be a Fredholm form and, by Proposition 2.3.2, its negative eigenspace will be commensurable with E-, so that iE- (a) is well defined. The relative Morse index is monotone with respect to the usual ordering of quadratic forms and it has the usual semi-continuity properties.

Proposition 2.3.3 Let a, r be Fredholm forms whose negative eigenspaces are commensurable with E- and such that a > r. Then iE- (a) :5 ZE- (r),

iB- (a) < iE- (r)

Let (an) be a sequence of Fredholm forms whose negative eigenspaces are commensurable with E-. Assume that (an) converges to or, meaning that


54

the corresponding self-adjoint operators Sn converge to S. Then a is a Fredholm form, its negative eigenspace is commensurable with E- and iE- (a) < lira inf iE- (an),

i(c) 1 lira Sup iE_ (an). n-soo

Let S, T be the self-adjoint realizations of a and r. Since

Proof.

S>T,

V-(S) fl [V+(T) (DKerT] = {0},

[V-(S) ®KerS] fl V+(T) = {0}.

Hence

iE- (a) - iE- (r) = dim(V (S), E-) - dim(V (T), E ) = dim (V - (S), V - (T)) = - dim[V'(S) ® Ker S] fl V - (T) < 0, and, similarly,

(a) - iE_ (r) = dim(V-(S) ® Ker S,V-(T) (D KerT) _ - dim V(S) fl [V-(T) ® Ker T] < 0. This proves that iE- and iE_ are order reversing. Now assume that (Sn) is a sequence of self-adjoint Fredholm bounded operators converging to S. Then S is a Flredholm operator, so or is a Ftedholm form. Let Q be the orthogonal projection onto Ker S, so that S f Q are self-adjoint and invertible. Since Q > 0, 2E- (Sn + Q)

_ nl

iE- (Sn + Q) = dun(V (S + Q), E) =1E- (S),

oof

limsupiE_(Sn)-< lim iE-(Sn - Q) = dim(V (S -Q),F%) _ _E- (S) n-+oo

0 We conclude this section with an obvious result about the relative Morse index of forms which split. The easy proof is left to the reader.


55

Proposition 2.3.4 Let a[-, -] = (S., -) be a Fredholm form, whose negative eigenspace is commensurable with a closed subspace E-. Assume that

E- =Ei ®...®Ek,

E=E1 S, --- eEk,

are orthogonal sums of closed subspaces. If the subspaces E1 are S-invariant then k

ZE-(a) = EiE, (OIEi). j-1

2.3.1

Further properties

Proposition 2.3.2 asserts that the positive eigenspaces of two self-adjoint operators whose difference is compact are commensurable. A sort of converse statement also holds.

Proposition 2.3.5 If S is an invertible self-adjoint operator with positive (negative) eigenspace E+ (E-) and E = W+ ® W-, where W+ (W-) is commensurable with E+ (E-), then there exists an invertible self-adjoint operator T whose positive (negative) eigenspace is W+ (W-) and such that S - T is compact. Proof. Let P+ and P- be the orthogonal projection onto E+ and E-; let Q+ and Q- be the orthogonal projection onto W+ and W-. Set

T:= Q+P+SP+Q+ + Q-P-SP-Q-. Then _T is a bounded self-adjoint operator, W+ and W- are T-invariant

and T > 0 on W+, t < 0 on W-. Since E+ and E- are commensurable to W+ and 44'-, the operators P+Q-,

P-Q+, Q+P

,

Q-P+,

are compact, so the operator

S - T = (Q++Q-)(P++P-)S(P++P-)(Q++Q-)-T is also compact. Therefore, T is a Fredholm operator, hence its kernel is finite dimensional and it must have the form KerT = Z+ ® Z-, where

Z+ C W+ and Z- C W-. Then the operator

T:=T+Pz+-PZsatisfies all the requirements.

0


56

If a is a continuous symmetric bilinear form on E and V C E is a subspace, the a-orthogonal complement of V is the closed subspace

V l° :={yE EIa[y,x]=0, bxE V}. The following facts will be used:

(i) (V +W)1° = V- n IV(ii) Ker S C V 1

(iii) if a is a Fredholm form, (V 1° )1O = V + KerS.

The proofs of (i) and (ii) are immediate. To prove (iii), notice that since S is invertible on (KerS)1, there exists a self-adjoint Fredhohn bounded operator R such that

RS=SR=P(KerS)1 From the identities W1°

= (SW)1 = RWl + KerS,

we get

(V1°)1° = ((SV)1)1° = RSV + KerS = V + KerS, proving (iii). It is interesting to deal with maximal subspaces W # V - (S) on which

a is strictly negative. Such subspaces need not be commensurable with V-(S), but they will always form a Fredholm pair together with V+(S). More precisely, the following statement holds.

Proposition 2.3.6 Let

a Fredholm form on E. Let V, W be closed subspaces of E such that V = W1°, a is positive on V and strictly negative on W. Then (W, V+ (S) ® KerS) is a Fredholm pair of index zero.

Before proceeding with the proof, it is worth noticing that, under the above assumptions, V + W is dense in E, but it may not be closed. Proof. Since a is strictly negative on W and positive on V+(S) KerS, w n (V+(S) ® KerS) = {0}. Since a is a Fredholm form, we can decompose every y E E as y = y++y- +y°. where yt E V (S), y° E Ker S, and

a[y]2 =

a[y+]2

+ a[y ]2 >_

lIy

112

- cII y 112,

for some c > 0. Since a is negative on W, we get lly+ll S clly ll.

dy E W.

(2.7)

2.4. Relative Morse index of critical points

57

Let (y,) C IF. (zn) C V' (S) be such that yn+xn - Z. Then yn +zn -> z+ yfz -a :- and y° -4 z°. By (2.7). IIy; - y,,;II < calyn - yrnll.

and so that is a Cauchy sequence and it must converge. Hence converge, so : E 11'+ V+ (S). proving that IV+ V+ (S) is closed. Since KerS is finite dimensional, also IV + (1-+(S) ::,- KerS) is closed. Since a is positive on t- and strictly negative on V- (S), by (1).

(11' + (1"(S)

KerS)]1^ = 1' n [I'-(S) -?- KerS] = ker S.

Passing to the a-orthogonal complements in the above identity and using (iii). we obtain it' + (1"+ (S)

so that (ti; I''(S)

2.4

KerS) = (KerS)1° = E.

Ker S) is a Fredholm pair of index zero.

C7

Relative Morse index of critical points

Let r be a critical point of a twice differentiable functional of the form Au):= 2 (So u. u) + b(ra).

(2.8)

where S° is a self-adjoins invertible operator and Vb : E -a E is a compact trap, meaning that it traps bounded sets into relatively compact sets. Then D2 f (x), the Hessian of f at z. defined to be the bounded self-adjoint operator such that (D' -'f (x) u, r) = d2 f (r) [u, r].

has the form D2 f (r) = So + D2b(.r). and the self-adjoint operator D2b(x) is compact. Indeed. the compact (nonlinear) map

Vb(z + f) - t'b(x) f

converges to the (linear) operator D2b(r) uniformly on hounded sets, as f -> 0. Therefore, cP f (x) is a Fredholm form. Let

E=E- - E" be the orthogonal splitting into the positive and the negative eigenspaces of S°. By Proposition 2.3.2, the negative eigenspace of D2f(X) is commensurable with E-, so we can give the following definition.


58

Definition 2.4.1 Let x be a critical point of a functional f of the form (2.8). The relative Morse index of x with respect to E- is the integer ME-(X):= iE- (d2f (x)) = dim(V (D2f (x)), E-). The relative large Morse index of x with respect to E- is the integer mE- (x) := iE- (d2f (x)) = dim(V (D2f (x)) (D Ker D2f(X) , E-).

2.5

Finite dimensional reductions

Throughout this section we will assume that the Hilbert space E is separable. As before, let So be an invertible self-adjoint operator. Set [So] := IS E C(E) I S - So is compact).

We would like to find the Morse index of a self-adjoint operator S E [So] looking at its finite dimensional reductions P,,SPP, where (P,,) is a sequence of finite dimensional orthogonal projections. Notice that this makes sense because PnSP,, is still a self-adjoint operator. In order to do this, we shall need the sequence (Pn) to be somehow compatible with So. Hence, we give the following definition, due to Chang, Liu and Liu [CLL97].

Definition 2.5.1 An approximation scheme with respect to So is a sequence (Pn) of finite dimensional orthogonal projections such that

(a) R(Pn) C R(Pm) for n < m;

(b) P - I strongly; (c) [Pn, So] := PnSo- SoPn -+ 0 in the operators' norm.

Remark 2.5.1 Property (a) is not assumed in Chang, Liu and Liu's original definition. Indeed, such a property will not be used in this section. However, it is by no means restrictive and it will be useful later on.

It is easy to build an approximation scheme (Pn) for So such that [Pn, So) = 0, when So admits a complete orthonormal sequence of eigenvectors. However, a self-adjoint operator in an infinite dimensional Hilbert space may not have any finite dimensional non-trivial invariant subspaces. Hence, in general there will be no finite dimensional orthogonal projection commuting with So. The existence of approximation schemes for a general So is proved by a perturbation argument.

Proposition 2.5.1 Approximation schemes (Pn) with respect to So exist.

2.5. Finite dimensional reductions

Proof.

59

First assume that E = L2(X, E,p), for some regular positive

finite measure space (X, E, p), and that So = L f is the multiplication operator L fg = fg by some f E L°°(X, E, p). Consider a sequence of finite measurable partitions X = U 1 X, such that:

(a) the partition (X,`% refines the partition (X7 )j: for every i E { 1, ... , kn+1 } there exists j E { 1, ... , kn} such that X; +1 C X7 ;

(b) {Xx; E n E N, j = 1.....k}, where Xx denotes the characteristic function of X, spans a dense subspace of L2(X, E, p). (c) (If++ - f IIL '(X.E,1) < 2-n, where fn

ax,, o7 E R.;

Let Pn be the orthogonal projection onto the finite dimensional space l X x; I j = 1, ... , kn } . By (a), R(Pn) C R(P,n) for n < m. By (b), P,, -. I strongly. By (c). L f, - L f in the operators' norm and, since [Pa, Lf.] = 0, we get that [P, , L f] - 0 in the operator norm. The general case follows because every bounded self-adjoint operator on a separable Hilbert space is unitary equivalent to a multiplication operator on L2(X, E, p) by some f E Lx(X. E. µ). for some regular positive finite measure space (X, E, p) (see Corollaries X.5.1 and X.5.5 in [DS63]).

D

The strong convergence Pn - I becomes a convergence in the operators'

norm when Pn is composed, on the left or on the right, with a compact operator. More precisely, we have the following result, whose easy proof is left to the reader.

Lemma 2.5.2 Let K be a compact subset of the set of compact operators. If (Pn) is a sequence of finite dimensional orthogonal projections which 0 and KP - 0 in the converges strongly to the identity, then operators' norm, uniformly for K E K. The above Lemma has the following consequence: if (P,) is an approximation scheme with respect to So, it is also an approximation scheme with respect to every invertible self-adjoint operator S1 E [So].

Proposition 2.5.3 Let (Pn) be an approximation scheme with respect to So. Let K be a compact subset of the set of compact self-adjoint operators such that So + K is invertible for each K E K. Then there exist b > 0 and a positive integer no such that, for n > no and K E K, we have (i)

I > b11Pnxll, for every x E E;

(ii) IIPPSoPnxI) >_ a11Pnxil, for every x E E;


60

NO iE- (SO + K) = i(PP(So + K)PP) - i(PPSoPJ. Proof. By Lemma 2.5.2 and property (c) of approximation schemes, [Pn, So + K] -+ 0 uniformly in K E A. Then the estimate IIPP(So + K)PnxII >- II(So + K)Pxll - II[Pn, So + K]IIIIPnxll >>

( II(So+ K)-'II 1

- II[Pn, So + K]II) IlPnxll

implies (i). Since we may assume that 0 E IC, (ii) follows. Set

S. = PnSoPn + (I - Pn)S0(I - Pn),

so that S, -+ So and S + PnKPn - So + K in the operators' norm. By Propositions 2.3.1 and 2.1.1, lim

lim [dim(V-(Sn+PnKPP),E-)-dim(V-(PnSoPP),E-)]

n-too

= dim(V-(So + K), E-) - dim(V-(So), E-) = dim(V-(So + K), E-), uniformly in K E K. Therefore, there exists no such that, for every n > no

andKE1C, dim(V-(Sn + PnKPP),V-(sn)) = dim(V-(So + K), E-).

(2.9)

The formulas

V-(Sn + PnKPn) fl V+(Sn) = V-(Pn(So + K)Pn) fl V+(PPSoPP), V+(Sn + PnKPn) n V (Sn) = V+(Pn(So + K)Pn) n V (PnSoPn), imply the identities

dim(V (Sn + PnKPn), V-(Sn)) = dimV-(Sn + PnKPn) f1 V+(Sn) -dimV+(Sn +PnKPn) fl If (Sn) = dim(V -(Pn(So + K)Pn), V -(PnSoPP)) = dimV-(Pn(S0 + K)Pn) - dimV-(P,,SOPP) = i(Pn(So + K)PP) - i(PnSoPn). Hence (iii) follows from (2.9).

The above proposition has the following immediate consequence.

0

2.5. Finite dimensional reductions

61

Theorem 2.5.4 Let So be a self-adjoint invertible operator with positive and negative eigenspaces E+ and E-. Let (Pn) be an approximation scheme with respect to So. Let S be a self-adjoint operator such that S - So is compact and let Q be the orthogonal projection onto Ker S. Then

iE- (S) = lira [i(Pn(S + Q)PP) - i(PnSoPP)]. n-+oo

(2.10)

iE-(S) = lim [i(Pn(S - Q)Pn) - i(P,,SoPn)]. n-40C

(2.11)

The above limits are uniform, for S varying in a compact set.

Formulas (2.10) and (2.11) are Chang, Liu and Liu's definition of the relative Morse index. The preceding theorem shows that the direct approach and the finite dimensional reduction method are equivalent. The following facts will be useful later on.

Proposition 2.5.5 Assume that (Pn) is an approximation scheme with respect to the invertible self-adjoint operator M. Let P+ and P- be the orthogonal projections onto the positive and the negative eigenspaces of M.

Let P and Pn be the orthogonal projections onto the positive and the negative eigenspaces of PnMP,,. Then

(i) P -* P+ and Pn - P- strongly; (ii) P+ Pn -r 0 and P- P1, -4 0 in the operators' norm. Proof.

Let

Mn = PPAfPn + (1- Pn)M(1- Pn). Then Mn is a self-adjoint operator and Afn - M = Pn[Af, Pn] + [Pn, M]Pn,

so that, by property (c) of approximation schemes, Mn -a Al in the operators' norm. Let Q+ and Q; be the orthogonal projections onto the positive and the negative eigenspaces of Afn. Then

Qn = XR+(Mn) -> XR+(M) = P+, Qn = XR- (Afn) -4 XR- (M) = P-, in the operators' norm. Therefore, statement (i) follows from the second property of approximation schemes and from the identities

P = PnQn , Pn = PnQn Statement (ii) follows from the identities

Pn = Qn Pn,

Pn = Qn Pn. X

62

2.6



The notions of commensurable subspaces and relative dimension were introduced by the author in [Abb97], with a slightly different formalism. More properties and simplified proofs were provided in [AbbOO], [AM00b], and [AM00c]. They provide the natural language to deal with indefinite quadratic forms. The term commensurable was suggested to the author by Joseph Bernstein. Fredholm pairs of subspaces of Banach spaces are used by Kato [Kat80] to prove classical results about Fredholm operators. Therefore, this concept should be considered more primitive than the notions of commensurable subspaces and relative dimension, some of whose properties are proved using classical results about the Fredholm index. All the results stated for self-adjoint Fredholm operators could be extended to hyperbolic operators, i.e. operators whose spectrum is disjoint from the imaginary axis (or, more generally, to operators whose essential spectrum is disjoint from the imaginary axis). Another way of interpreting the relative Morse index is by looking at the so called spectral flow. In our case, one would be interested in paths of operators of the form So + K(t), with K(t) compact. Such an approach can be found in [AvdV99] and in [AM00b]. See also [Sch93] for the finite dimensional case, [RS95] for paths of unbounded operators with compact resolvent and [AM00a] for arbitrary paths of bounded operators on a Hilbert space.

Chapter 3

Functional setting It has been known for a long time that the T-periodic solutions of a Hamiltonian system

I p' _ q' =

H(p,q)

are the stationary curves of the action functional

fo7[p(t) q'(t) o n the space of smooth T-periodic loops. To use this variational principle effectively, it is necessary to extend the above functional on a suitable Sobolev

space. Critical points of such a functional will be classical T-periodic solutions of the Hamiltonian system. Although they have an infinite Morse index and co-index, the concepts introduced in Chapter 2 will allow us to introduce a finite relative Morse index. The main result will be that such an index coincides with the Maslov index of the linearized system along the solution.

Although in this book we will not deal with second order Lagrangian systems, their appropriate functional setting and questions concerning their Morse index will be among the topics of this chapter.

3.1

Fractional Sobolev spaces

In this section, we recall the definition and the basic properties of fractional

Sobolev spaces. Let S' := R/Z be the circle, parameterized on [0, 1]. If 63

Chapter 3. Functional setting

64

a > 0 is a real number, set

H'(S';Rm):= {uEL2(SI:Rm)IEIkI2'IukI2 <xI kEZ l where (uk) C C' is the sequence of Fourier coefficients of u: uke2aikr

u(t) _

u_ k =Il k

kEZ

Then H'(S';Rm) is a real Hilbert space with the inner product (u, v)'

1k12,(uk. i k),

(IOe io) + 27r

kEZ

denotes the standard Hermitian product of Cm. The norm will be denoted by 11.11' := Smooth functions are contained in H' for every s > 0 and they form a dense subspace: indeed, trigonometric polynomials form a dense subspace. When s = 0, H°(S' ; Rm) is nothing else but the space of square integrable functions where

related to

from SI to R"', and Ilullo is equivalent to the usual norm 1

1

IIUIIL2

(J Iu(t)12 dt) 0

When s is a positive integer, H'(S';R") is the space of functions whose weak derivatives up to the s-th are square integrable. The norm II II' is equivalent to the norm

' IIuIIH :=

IID'uI1 2 i-o

The following results are standard in the theory of Sobolev spaces (see [Ada751), but, since it is not so easy to extract the statements we need from the general theory, we provide explicit proofs.

Theorem 3.1.1 The following embeddings are compact.

ifs>r>0,

(i)

(ii) H'(SI; R-) --? CQ(S'; Rm), if s > 2 + q and q is a non-negative integer;

(iii) H I (S'; Rm) y 17(S' ; Rm), for every p E [1, +oo[.

3.1. Fractional Sobolev spaces

65

(i) By definition. llullr < 11uli, if s > r, so the natural em-

Proof.

bedding I : H" --r H' is continuous. Let P : H' - Hr be the operator with finite rank

^ Uke2rikt

Pnu =

I kk a,

iE-(a)

iE-(an) = µ(Bn),

for n large. Since (B,,) converges uniformly to B,

p(B) < lim inf p(Bn)

iE- (a),

0

concluding the proof.

As a first simple application of the above theorem, we can show that the Maslov index of a strictly convex Hamiltonian system is not less than N.

Proposition 3.2.3 Assume that the symmetric matrices B(t) are positive for every t E [0,11, and that B(to) is strictly positive for some to. Then

p(B) > N. Proof.

Under the above assumptions, the quadratic form as[z12 =

-J

1

Jz'(t) z(t) dt - / B(t)z(t) - z(t) dt

0

o

is strictly negative on

E-(P RN= aEElz(t)_Ee21rkJ!zk k 1. The action functional determined by the Hamiltonian H is defined as eH(z) := J 1 p(t) q'(t) dt - / H(t, p(t), q(t)) dt 1

0

0

_

-12

/ 1 Jz'(t) z(t) dt - / 1 H(t, z(t)) dt. Jo

o

It is easy to see that the stationary curves of the action functional on the space of smooth loops in R2N are exactly the one-periodic solutions of system (3.16). In order to apply variational methods to the functional eH, it is convenient to extend it on the Sobolev space E = Hi(S1;R2N). Indeed, we have seen in section 3.2 that, in this space, the leading part of the functional -1 ' Jz'(t) - z(t) dt Jo

is a Fredholm form. However, since curves in Hi are not bounded, H(t, z(t))

needs not be integrable. Therefore, the HI formulation is possible only when H satisfies some mild growth conditions.


76

Assumption 3.3.1 The Hamiltonian H : S1 x R2N - R is twice continuously differentiable and there exist constants c, p > 0 such that 2

z2 H(t,

z) dim V(T), on which r is strictly negative, which is impossible. Set

V+:= {z EV Ia[z,w]=0,V1eEV-}. The maximality of V- implies that a[z]2 > 0 for every z E V+, proving our claim. The restriction of or to W is ajW[z]2 = 0,[(p,0)]2 = -

J

1

;

(t)-'p(t)

. p(t) dt.

Since the matrices X(t)-1 are strictly positive, a is strictly negative on IV. Since W and V- are a-orthogonal, a is strictly negative on W := W &3 ti''-, which is closed because V- is finite dimensional. Moreover, IV'- = V+ and Proposition 2.3.6 implies that (W,V+(S) ® KerS) is a F edholm pair of index zero. Then (W, V+(S) ® Ker S) is a Fredholm pair of index

ind (W, V+ (S) ® Ker S) = - dim V- = -1(r). Since V+ (S) ® Ker S and E+ are commensurable, Proposition 2.2.1 implies

that ind (W, E+) = ind (W, V+ (S) 9 Ker S) + dim(E+, V+ (S) ® Ker S)

= -i(r) + dim(V-(S), E-). Using the explicit Fourier representation for E+ (3.20), it is easy to check that ind (W, E+) = 0, so the above identity gives iE- (a) = dim(V (S), E-) = i(r), concluding the proof.

Together with Theorem 3.2.2, the above theorem has the following immediate consequence.

Corollary 3.4.2 Let L be a periodic quadratic Lagrangian satisfying the Legendre positivity condition. Let p(B) and v(B) be the Maslov index and the nullity of the corresponding first order Hamiltonian system (3.22). Then

i(r) = µ(B),

n(r) = v(B).

3.5. Nonlinear Lagrangian systems

3.5

83

Nonlinear Lagrangian systems

A nonlinear periodic Lagrangian system takes the form d dt 0

L(t,q(t),q'(t)) = a L(t,q(t),q(t)),

(3.23)

where L, the Lagrangian, is a smooth function in (t, q, v) E S' x R' x RN. Important examples are provided by Lagrangian of the form L(t, q, v) = 2 Iv12

- V(t, q),

which give rise to the second order systems

q `(t) _ - qV (t, q(t)). If 82 L(t, q, v) is invertible for every (t q, v) E S' x RN x RN, it is possible to pass to the Hamiltonian formulation by means of a Legendre transform. Indeed, the invertibility of i2 L allows us to apply the global implicit function theorem, obtaining a smooth function v : S' X R2N - RN such that D,.L(t, q, v(t, p, q)) = p,

Vp E RN.

Then the Legendre transform of L is the Hamiltonian

H(t,p,q) := p- v(t,p,q) - L(t,q,v(t,p,q)). The change of variables q(t) = v(t, q(t), p(t)) makes system (3.23) equivalent to the Hamiltonian system determined by H:

f p'(t) _ q'(t) =

(3.24)

Therefore, we can associate a one-periodic solution q of the Lagrangian system (3.23) with all the invariants of the corresponding solution (p, q) of the Hamiltonian system (3.24). Definition 3.5.1 The periodic solution q of (3.23) is non-degenerate if the corresponding solution 2 = (a3, q) of (3.24) is non-degenerate. The nullity of q is the nullity of 2: v(q) := v(2). The Maslov index of q is the Masloa index of 2: µ(q) = p(2). The mean winding number of q is the mean winding number of 2: r(q) := r(2).


84

Linearization of (3.23) along q yields

d

j802L

dt

q

02L

8v&V2

OQL(t,9,4)q+

q (3.25)

which is a linear periodic Lagrangian system. It is easy to see that the quadratic Lagrangian giving the above system generates, by a Legendre transform, the quadratic Hamiltonian corresponding to the linearization of (3.24) along (p, q). In other words, the operation of linearization commutes with the Legendre transform. Again, if L does not depend on t, the function q' is a non-periodic solution of (3.25). Therefore, a non-constant solution of an autonomous system is always degenerate: v(q) > 1. The Euler functional determined by the Lagrangian L is defined as fl, (q)

fo

1

L (t,q(t).q(t))dt.

It is easy to see that the stationary curves of the Euler functional, on the space of smooth loops. are exactly the one-periodic solutions of system (3.23). Under reasonable assumptions on the growth of L in the v variables (we do not need to make this more precise here), fL extends to a smooth functional on the space F := H' (S' : RN), a convenient choice in order to use variational methods. Then critical points of fl, in F are the classical one-periodic solutions of (3.23). Let q be such a solution and let 1 82L

82L

Lq(t, q v) := 2 8ij2 (t' 4, 4')V ' L + 8t-8q (t, 4, 4)q v +

182L 2

aqs (t., 4, 4')q ' q

be the quadratic Lagrangian giving the linear system (3.25). Then dzfL(4)[q]2 = 2 fot L

which is exactly the quadratic form r corresponding to the quadratic Lagrangian L4, introduced in section 3.4. So the kernel of the second differential of fL consists of the one-periodic solutions of the linearized system (3.25), hence v(q) = n(d2 fL(q)). The Legendre positivity condition of Definition 3.4.1 extends to arbitrary Lagrangians in an obvious way. Definition 3.5.2 A Lagrangian L is said to to satisfy the Legendre positivity condition if 2 L(t,q,t') is a strictly positive matrix, for every (t, q, v). Therefore, the one-periodic solutions of a Lagrangian system satisfying the Legendre positivity condition have finite Morse index, as critical points of fL, and Theorem 3.4.2 can be restated in the following way.

3.6. Some bibliography and further remarks

85

Corollary 3.5.1 Assume that 4 is a non-degenerate one periodic solution of the Lagrangian system (3.23), where the Lagrangian satisfies the Legendre

positivity condition. The Morse index of 4 as a critical point of fL on F coincides with the Maslov index of 4:

m(4; IL) = P(4)

3.6


The fundamental idea of looking at the action functional of a Hamiltonian system in a H 2 setting is due to Rabinowitz [Rab78a]. Conley and Zehnder's theorem 3.2.2 and its Lagrangian counterpart 3.4.1 describe relations between the index of some form in a functional space and a topological invariant, namely the Maslov index. This kind of

result has a long history in mathematics, starting with the Morse index theorem for geodesic [Mor47] (see also [Sma65b]), Sturm oscillation theorem (see Chapter 8 in [CL55], [Edw64]), Bott's iteration theory of closed geodesics [Bot56], passing to the already mentioned paper by Duistermaat [Dui76] (see also [CD77]), and arriving to Ekeland's index theory for convex Hamiltonian systems [Eke86. Eke9O], and the connections between the Maslov index and the Fredholm index of suitable Cauchy-Riemann operators used in Floer homology [SZ92, RS95]. A comprehensive introduction

to this subject can be found in the lecture notes by Piccione and Tausk [PTOO] (see also [PT99]).

Chapter 4

S uperquadrat is Hamiltonians A superquadratic Hamiltonian is a function H(z) which behaves as IzI°, for

some 8 > 2, when z lies outside from a large ball in R2N. Although superquadratic Hamiltonians may not seem physically interesting (in particular, it is the superquadratic growth in the p variables which seems unphysical) they constitute important mathematical examples. Their main feature is that the corresponding action functional satisfies a useful compactness condition, the Palais-Smale condition. This fact enabled Rabinowitz to prove the first variational results about existence of periodic solutions of such systems [Rab78a). Moreover. many other problems, such as finding closed characteristics on energy surfaces or computing symplectic capacities. can be handled by introducing suitable superquadratic Hamiltonians. This chapter starts with a review of some tools in critical point theory, namely Morse theory and linking theorems, which will be useful also in the next chapters.

4.1 4.1.1

Abstract critical point theory The (PS) and (PS)* condition

Let M be a Hilbert manifold endowed with a Riemannian structure g. Assume that M is a complete metric space with the distance induced by g. Let f E C' (At). The Riemannian structure allows us to associate a vector field, the gradient of f, denoted by V f , to the the differential of f : gZ(V f (x), u) = df (x)[u], 87

Vu E TTM.

Chapter 4. Superquadratic. Hamiltonians

88

We will always use the standard notations

f a := {x E U I f (x) < al,

fa := {x E Af I f (x) > al,

f.':= fa n f b.

Denote by Crit(f) :_ {x E Al I df (x) = 01 the set of critical points of f . A real number a is a critical value if there exists x E Crit(f) such that f (x) = a, otherwise it is a regular value. We recall the well-known Palais-Smale condition.

Definition 4.1.1 Let J C R := R U {±x}. The functional f satisfies (PS) on J if every sequence C Al such that f (x,,) -+ c E J and V f (zn) -> 0 is compact. Sequences satisfying the above assumptions are called (PS) sequences. If J = R in the above definition, we will simply say that f satisfies (PS).

If f satisfies (PS) on J and Jo C J is a compact set. then Crit(f) n Jo is compact. In particular, since R is compact, the whole set of critical points of f is compact when f satisfies (PS) on R. When dealing with strongly indefinite functionals, we will often find it convenient to restrict the functional to a finite dimensional subspace. obtain

the existence of constrained critical points, then try to take a limit by choosing larger and larger subspaces. A stronger version of (PS) is needed in order for this approach to work. Assume that f E Ct (E), where E is a separable Hilbert space. Let (Pa) be a sequence of orthogonal projections with finite rank such that

(a) setting E := R(P,,), we have E C E,,, if n
A, VC

x

E(-1)k-°nn9(fa) _ 9=0

E(-1)k-grankH4(f6, f°;G). 9=0

92

Chapter 4. Superquadratic. Hamiltonians

where

mo(fa) = # {x E Crit(f) n fn I m(x) = q} . In (4.4), we are using the convention A'° = 0: this is related to the fact that points with infinite Morse index do not change the homotopy type of sublevels, because the infinite dimensional ball is retractable onto its boundary. Notice that the Morse relations imply that, when the assumptions of the above theorem are satisfied, HQ (f b, f°; G) is finitely generated for every q and every group G, and vanishes for q large enough. The polynomial Q may depend on G. In particular, the Morse relations imply the existence of a critical point of Morse index q, whenever HQ (f b, p: G) 0 0. It is useful to drop the assumption on the critical points being non-degenerate in the latter statement. Recall that a C' map is called a Fredholm map if its differential at every point is a Fredholm operator.

Theorem 4.1.2 Let f E C2 (:11) and let a < b be regular values of f . Assume that f satisfies (PS) on [a, b] and that V f is a Fredholm map. If Hq(f b, f °; G) # 0 then there exists a critical point r. E fa such that

m(r) < q:5 m'(.r). This theorem can be proven using Marino and Prodi's method of finding approximating functionals with only non-degenerate critical points [MP75]. Another proof is based on Benci's generalization of Conley theory (see Theorem 6.10 in [Ben9l]). Conley theory is a generalization of Morse theory. which allows us to work with flows which are more general than the negative gradient flow (see [Con78, CZ84, Sa185]). Ry bakowski and Benci have provided infinite dimensional generalizations of such a theory (see [Ryb87], [Ben9I] and [BG94]). Although we are not interested in flows other than gradient flows, we will find it convenient to borrow some concepts from Conley theory, so we give an extremely simplified account of such a the-

ory. stating just those results which will be useful later on. Again, finite dimensional statements would be enough for our needs. Since V f is a C' vector field, the following Cauchy problem d _ Vf(rl(t,x)) dt q (t ,,r) r1(0, x) = r,

1 + I[Gf (n(t, .r)) II '

defines a global flow q : R x V - Al. We will refer to q as to the negative gradient flow of f, although strictly speaking, this is just a time reparameterizatiou of the negative gradient flow, which may not be defined for every time t. If U C M. the maximal invariant set contained in U is the set 1(G) := {x E U I t1(R x {r}) C U).

4.1. Abstract critical point theory

93

Definition 4.1.3 A pair of closed subsets (U, Y), Y C U C M, is an index pair for f if the following conditions hold (i) f (U \ Y) is bounded;

(ii) I(U \ Y) C int(U \ Y); (iii) Y is positively invariant with respect to U: if x E Y and 77([0, t] x {x}) C U then ,([0,t] x {x}) C Y;

(iv) Y is an exit set for U: if x E U and 77(R+ x {x}) 0 U, there exists to > 0 such that 77([0, to] x {x}) C U and rl(to, x) E Y.

Here is a consequence of the Morse-Conley relations (see Theorem 3.3 in [CZ84], or Theorem 4.9 in [BG94]).

Theorem 4.1.3 Let f E C2 (M) and let (U, Y) be an index pair for f . Assume that f satisfies (PS) on Y (U \ Y) and that all its critical points in U \ Y are non-degenerate. Then there exists a polynomial Q with nonnegative integer coefficients such that the following identity holds:

E

Ar" (x) = P(U, Y; G) + (1 + \)Q(A).

(4.5)

zECrit(f)n(U\Y)

Since f is not increasing along i and strictly decreasing away from critical points, (f b, f a) is an index pair, whenever a < b are regular values. Hence the classical Morse relations are actually a special instance of the above theorem. The following result generalizes Theorem 4.1.2 (see Theorem 6.10 in [Ben9l], or Theorem 5.10 in [BG94]).

Theorem 4.1.4 Let f E CZ(M) be such that V f is a Fredholm map, and let (U, Y) be an index pair for f . Assume that f satisfies (PS) on f (U \ Y). If Hq (U, Y; G) 34 0 then there exists a critical point x E U \ Y such that

m(x) < q < m*(x). We recall that the cup-length of a topological pair (U, Y), denoted by CL(U, Y), is the maximum number m such that there exists cohomology classes aj E H* (U, Y), j = 0, ... , m, such that deg aj > 0 for j > 0 and ao U U an 0. When the critical points are not assumed to be nondegenerate, their number can be estimated in terms of the cup-length.

Theorem 4.1.5 Let f E C2(M) and let (U, Y) be an index pair for f . Assume that f satisfies (PS) on f (U \ Y). Then f has at least CL(U,Y)+1 critical points in U \ Y.

94

Chapter 4. Superquadratic Hamiltonians

Finally, we will state the Morse relations for functions whose critical set consists of non-degenerate critical manifolds, in the sense of Bott [Bot54].

Definition 4.1.4 Let Ko be a compact connected isolated subset of Crit(f ).

We say that Ko is a non-degenerate critical manifold if it is a finite dimensional manifold and, for every x E Ko, the restriction of DI f (x) to the orthogonal complement of TTKo is invertible. The Morse index of K0, m(Ko), is the Morse index of some point x E Ko (it is easy to see that such a number does not depend on x E Ko). Notice that, if Ko is a manifold contained in Crit(f ), D2 f (x) always vanishes on TTKo, so being a non-degenerate critical manifold actually means that the kernel of DI f (x) is not larger than is strictly necessary and that zero is an isolated point in the spectrum of D2 f (x). If Ko = {x}, we are actually asking that x is a non-degenerate critical point. Here are the Morse relations for functions with non-degenerate critical manifolds, as proved by Bott [Bot54] (see also section 10.3 in [MW89]).

Theorem 4.1.6 Let f E C2(M) and let (U, Y) be an index pair for f . Assume that f satisfies (PS) on f (U \ Y) and that Crit(f) fl (U \ Y) consists of non-degenerate critical manifolds K1,. .. , Kr.. Then there exists a polynomial Q with non-negative integer coefficients such that the following identity holds: E,\m(Kj)P(Kj; Z2) = P(U,Y; Z2) + (1 + \)Q(A)j=I

If one wants to use more general groups than Z2, the above Morse relations would look more complicated. Indeed, on the left-hand side one would have to replace P(KK; Z2)Am(Ki) with the Poincare polynomial of an

m(Kj)-dimensional vector bundle on Kj with respect to the same bundle minus the zero-section. The relative homology of such an object can be computed using Thom isomorphisms, and it equals the homology of Kj, shifted by m(Kj) when such a bundle is trivial or when G = Z2. Closely related to the above theorem is equivariant Morse theory (see Chang's book [Cha93], section 1.7).

4.1.3

Linking theorems and Morse index estimates

Linking theorems provide a simple but extremely powerful method to prove existence of critical points. They were introduced by Rabinowitz [Rab78b],

Benci [Ben80] and Ni [Ni80], as generalizations of Ambrosetti and Rabinowitz's mountain pass theorem [AR73]. Our approach differs slightly from these pioneering works, for reasons which will be explained shortly.

4.1. Abstract critical point theory

95

Definition 4.1.5 Let Q be a closed q-dimensional ball topologically embedded into M and let S C Al be a closed subset such that 8Q n S = 0. We say that 8Q and S homologically link if 8Q is the support of a non-vanishing homology class in HQ_t(AI \ S).

Equivalently, for every singular q-chain t: in Al such that I8t1 = 8Q, we must have It l n S i4 0. Here Itl denotes the support of the chain 1;. In particular, Q n S i4 0. because Q itself is the support of a q-chain (in this case, just a q-simplex) whose boundary has support 8Q. A useful example of a homologically linking pair is built as follows. Let

Al = E be a Hilbert space and let E = li'+ W , where dim 61 = q -1. Set S := 8Bp n W+, the sphere of radius p in W+. Let e be a unit vector in li'+ and set Q := (Be n W-) e [0.r]e, the cylinder based on the closed ball of radius s in TV-, of direction e and height r. Then it is easy to prove

that 8Q and S homologically link, provided r > p. Indeed, let P be the projection onto IV- with kernel W+ and consider the continuous map

a : E 3 z - Px+II(I-P)xIIeE 4i--E) Re. Then 0 maps E \ S into (W- a Re) \ {pe} and fixes the points of 8Q. If, by contradiction, [8Q] vanishes in Ha_I(E \ S), then 0.[8Q] = [8Q] would vanish in HQ_1((IS'- O Re) \ {pe}). However, this is impossible because Hq_1(Sq-1) 0 0 and the embedding

Sq-1 --OQ- (IW'- 0Re)\{pe}=Rq\10} induces an isomorphism of homology groups.

Theorem 4.1.7 Let Q C Al be a topologically embedded closed q-dimensional

ball and let S C Al be a closed subset such that 8Q n s = 0. Assume that 8Q and S homologically link. Let f E C(M) be a function with F-edholm gradient such that (i) suP8Q f < inf s f ; (ii) f satisfies (PS) on some open interval containing [inf s f, SUPQ 11. Then, if r denotes the set of all q-chains in M whose boundary has support 8Q, the number

c:= inf sup f tEr IEI

belongs to [infs f, supQ f] and is a critical value of f . Moreover: f has a critical point -+ such that f (:i) = c and the following estimate on the Morse index of a holds

m(s) < q:5 m*(i).


96

We shall prove this theorem under the further assumption that the regular values of f are dense. If dim M = m < oo, Sard's theorem implies

that this assumption holds, provided f E Cm(M). The result for f E C2 (M) then follows by a simple perturbation argument. See [Ben9l] for a proof in the general infinite dimensional case. We remark that we will use this theorem only with a finite dimensional Al. Since Q itself defines an element of r, c < supQ f . Since 8Q Proof. and S homologically link, the support of any chain in F must meet S. so c > infs f. Hence

isf f < c<supf. Q

Let a and b be regular values of f such that

supf supQ f be such that f satisfies (PS)' on [a, 0). If A C E, denote by N, (A) := {x E E I dist (x, A) < E} the Eneighborhood of A. Since Vb is compact, f is Lipschitz on bounded sets. so there exists E > 0 such that sup f < a,

inf f > a,

N.(S)

N.(8Q)

sup f < 3.

(4.8)

N (Q)

We claim that S,, C Ne (S) for n large enough. Indeed, if x E S,,, then (P/IIP+xlI)P+x E S and in view of Proposition 2.5.5 (ii)

IIx

III'pxll P+xII

IIx - PtxIl

+IIP+x

IIP+zIIP*2II

= 11P --T11 + IIP+xII - PI = lip-P.+-T11 + I11P+xII

m*(xn; fn) - dim E, > 1. D2f (x), the semicontinuity properties of the relative Since D2f Morse index (Proposition 2.3.3) imply that

iw- (D2f (x)) < 1 < i;v- (D2f (x)). Then the relative Morse index of x with respect to E- satisfies

mE- (x) =1E- (D2f (x)) = iw- (D2f (x)) + iE- (T) < 1 + dim(6S'-, E-) < iK,-(D2f (z)) + iE- (T) = iE- (D2f(x)) = mE- (x).

Since a and 0 were arbitrary close to infs f and supQ f, taking a further limit as in the proof of Theorem 4.1.7, we obtain the existence of a critical

point i such that f (z) E [infs f, supQ f] and

ME- (z) < dim(W-, E-) + 1 < mE- (x).

0 Theorem 4.1.9 could be generalized in several ways: more general functionals could be considered, as well as more general sets S and Q. Moreover. it seems reasonable that the above statement should hold also when f satisfies just the (PS) condition and E is not necessarily separable: one should either try to generalize the proofs of Lazer and Solimini [LS88j, or to use some generalized cohomology (see [Szu92, KS97, Abb97, GIP991) and then mimic the proof of Theorem 4.1.7. However, such a generality is not needed for the applications we will give.

4.2

Superquadratic Hamiltonians

Consider the autonomous Hamiltonian system

z'(t) = JVH(z(t)). where H E C2(R21v) satisfies the following conditions:

(hl) H > 0; (h2) H(0) = 0, dH(0) = 0 and dH(0) = 0;

(4.10)

4.2. Superquadratic Hamiltonians

101

(h3) there exist constants 0 > 2 and R > 0 such that

0 < OH(z) < z VH(z),

if 4z) > R.

These conditions were introduced by Rabinowitz, who used them to prove the existence of T-periodic solutions of (4.10) for every T > 0.

Theorem 4.2.1 (Rabinowitz [Rab78a]) Assume that H E C2(R2!`) satisfies (hl), (h2) and (h3). Then system (4.10) has at least one T-periodic solution z, for every T > 0. In this statement, it is not required that T is the minimal period of the solution. The proof of the above theorem can be found also in Rabinowitz's book [Rab86]. Theorem 6.10. Since (h3) forces H to be coercive, as we are going to show in a moment, (hl) and (h2) essentially require that the global minimum of H is achieved in some point with vanishing second differential: by a translation, we can assume that it is achieved in zero and, by adding a constant, that its value is zero. Actually, assumptions (hl) and (h2) could be eliminated (see [Rab83]).

The important assumption here is (h3). If zo E R2N, ]zol = 1, set yp(s) := H(szo). so that (h3) implies 0 < 8 (s) < s 0 and liml,l-.. H(z) = +oc; (h2') H(0) = 0, dH(O) = 0 and d2H(0) = 0; (h3') there exist c > 0 and a function 9(s) which diverges for s -+ +oc such that

D2H(z) > -cI + 0(Jzl)Px. where Px is the orthogonal projection onto a J-invariant plane X C

R2h.

Then, for every T > 0 the Hamiltonian system (4.10) has a nonconstant T-periodic solution z. Furthermore, the Maslov index of z satisfies µr(2) < N + 15 JUT (1) + PT(2).

Notice that, since J2 = -I. z and Jz span a J-invariant plane, for every z E R2N. The idea of the proof is simple: we can modify H outside from

a large ball so that (hl'), (h2'), (h3') still hold and H is superquadratic in the sense of Rabinowitz. Then we find a solution 2 using the Linking


103

Theorem 4.1.9, which gives also estimates on its relative Morse index, i.e. on its Maslov index. By these estimates, 2 cannot belong to the region where H was modified, because (h3') and the energy conservation law imply that solutions with large L°° norm must have large Maslov index. Therefore, z is a solution of the original problem. Now we come to details. Proof. It is enough to prove the existence of a one-periodic solution. Indeed, making the change of variables t' = It, equation (4.10) becomes

4

dz

= TJVH(z),

ctt'

and the Hamiltonian TH satisfies conditions (hl'), (h2') and (h3'), with c replaced by Tc and 0(s) replaced by TO(s). By (h3'), for every a > 0 we can find R = RQ > 0 such that

D2H(z) > -ci + aPX,

if Izi > R.

(4.13)

We will specify later how large a must be. By (hl') we can find S > R such

that max H < BR

inf

R2N\Bs

H.

It is possible to build a non-negative function K E C2(R2N) such that:

(kl) K(z) = H(z) for z E BS; (k2) K(z) = M(Izj4 - S4) for I z I > S + 2 (M is a positive constant); (k3) maxBR K< infft2N\B5 K;

(k4) D2K(z) > -cI + aPx for jzi > R. To build such a K, choose a smooth non-increasing function X : R H R such that X(s) = 1 for s < S + 1 and X(s) = 0 for s > S + 2, and set K(z) := X(IzI)H(z) + M max{(Iz14 - S4), 0}. Obviously K is positive, of class C2 and satisfies (kl), (k2). It also satisfies

(k3) if M is large enough, because K > H in BS+1 \ BS and K > M outside BS+1. Finally, M can be chosen so large that K satisfies (k4), because D2K > D2H in BS+1 \ Bs and D2K(z) > -kI + 12M(S + 1)2I,

for IzI > S + 1,

where

k := sup ID2(X(Iz1)H(z))I. zER2N


104

The Hamiltonian K satisfies the growth assumption 3.3.1, so the action functional

eK(z) _

-1

/ Jz'(t) z(t) dt - j K(z(t)) dt

is twice continuously differentiable on the Hilbert space E = HI (S'; R2^') For simplicity, we will denote the norm and the inner product of E as II ' II The critical points of eK are one-periodic solution of system and

z'(t) = JVK(z). Moreover, as we have seen in section 3.3, eK has the form eK(z) = 2 (Soz, z) + b(z),

where S° is self-adjoint and invertible, and b has compact gradient. Let E+ and E- be the positive and negative eigenspaces of So, as in section 3.2. Set

En

t

1

e2akJtzk

kn

I zk E R2^'

and let Pn E G(E) be the orthogonal projection onto En. Then So commutes with Pn, so (Pn) is an approximation scheme with respect to So. Claim 1. The functional ell satisfies the (PS)' condition with respect to (Pn). Indeed, let (zn) C E be a (PS)' sequence: zn E En, IeK(zn)I < co and PnVeK(zn) -1 0. Then deK(zn)[Zn] = o(IIznII), so, if n is large, Co + IIznII 1 eK(zn) - deK(zn)[zn] = fo 1

-() [viczn

-

zn - K(zn)1 dt.

By (k2 ),

IVK(z) z - K(z) >

MIz14

- c1,

Vz E R2N.

Here and in the following cl, c2.... denote suitable positive constants. Hence CO + IIZnII > MIIznIIL - Cl.

(4.14)

We can write every z E E as z = z+ + z- + z° where z° is constant, while z+ E E+ and z- E E- have vanishing mean. Then (4.14) implies CO + IIznII >t C2IIznIIL2 - Cl > C21zn14 - cl,

4.2. Superquadratic Hanliltonians

105

which can be written as IZ°I < C3(1 + IIZnIl4)

(4.15)

Since Z' E En, dek (zn)[z,+, j = o(IIzn II), so, if n is large u

I(S0z

.

n) -

Vh (z,,) ' zn dt1 = deK(Zn)lz+'}i j < IIzn II 0

Therefore, by Holder inequality, since So = I on E+,

I1 I14n l1 = (Sozn, zn) < < II4+IIL4

fo

U

VK(zn) ' zn dtI + (Izn I(

IV

(Zn)Idt) +IIZ+II

Since E continuously embeds into L4 (Theorem 3.1.1). by the estima IVK(z)I < c4(1 + (z13) we deduce that

-

I1=+112 < C5(1 + IIznIIL4)Ilzn II 11

Ilzn II < c5(1 + IIzfflL4) p > 0 ands > 0 such that supaQ f < 0
0 we can find d > 0 such that K(z) < eIZI2 for Izi < S. Since K grows as a fourth power, there exists C(c) such that K(z) < elzlz + C(e)IzI4,

`dz E R2N.

Then 1

K(z) dt

4

IIZII2 - a1C(e)Ilz1I4 = 1pz. 8

If z E W- we can decompose it as z = z- + z°, where z° is constant and z- E E- has vanishing mean. Since K(z) > MIzI4 - a2, we have the chain of inequalities

r 0

r

K(z+Ae)dt>_MJz+Ael4dt-a2>MIz+e12dt) -a2 0 (L' = M (Iz°I2 + IIz IILz + A2Ileili 2)2 - a2 > a3(Iz°I4 + A4) - a2.

Then,

e.K(z + )e) =
p so large that cp(.\) :_

1 A2

- a3\4 + a2 < 0,

for all A > r. Set

L:= max p(A). AEi0,r]


107

Let s > 0 be such that 2jjz-jj2 +a3jz°j4 > L,

if z E W` and jjzjj > s. Therefore, if jjzjj = s and 0 < A < r, e.K(z + Ae) < L - I llz`112 - a3jz°j4 < 0. Moreover,

eK(z + re) _ ;p(r)

- 2jjz

112

- a3jz°j4 < 0.

Finally, eK(z) < 0 if z E W', because K < 0. Thus we have proved that eK < 0 on 8Q and Claim 2 holds. By Claims 1 and 2, Theorem 4.1.9 can be applied and we get the existence of a critical point 2 such that eK (2) > infs eK > 0, with index estimates

mE- (z) < dim(W , E`) + 1 = N + 1 < mE_ (z).

(4.18)

By Theorem 3.2.2,

p(z) < N + 1 < µ(z) + v(2)Moreover, eK (2) > 0, so z cannot be a constant. We want to show that jz(t)I < S for every t, so that 2 is a solution of equation (4.10). By the energy conservation law for autonomous Samiltonian systems, K(z(t)) is a constant. Therefore, by (k3), it is enough to show that z(to) E BR for some to. Assuming the contrary, that is 2(t) > R for every t, using (k4) we will get an estimate for the relative Morse index of z which violates (4.18). In fact, by (k4), (D2eK(z)z, z)

=-

r!o J t 1

I0

Jz'(t) z(t) dt - fo i D2K(z(t))z(t) z(t)dt

Jz'(t) z(t) dt -

Io

I (-cI

+ aPX)z(t) . z(t) dt.

The latter quadratic form. which will be denoted by oa, is associated to the linear autonomous Hamiltonian system

w'(t) = JBw(t),

where B := -cI + aPV.

Since X is a J-invariant plane, the matrix JB has eigenvalues ±ci, each with multiplicity N-1, and ±(a-c)i, each with multiplicity one. Moreover,

-d.....-ci.(a-c)i


108

are the Krein-positive eigenvalues. We may assume that c V 2iZ and a - c 0 2,xZ, so, by Theorem 1.5.1, the above linear system has Maslov index µ(B) _ (1V - 1)[[-2c]J + [(2(a - c))J.

We can choose a so large that the above expression is not less than N + 2. Since the relative Morse index is monotone with respect to the usual ordering of self-adjoint operators (Proposition 2.3.3) and since it coincides with the Maslov index, mE- (z) = iE- (D2ex(z)) >- iE- (us) = p(B) > N + 2. The above inequality contradicts (4.18), concluding the proof.

o

A similar argument proves that, if (h3') is replaced by:

(h3") there exist c > 0, R > 0 and 0 > Nc such that

D2H(z)>-c1+9PX, forjzj>R. one gets the existence of a T-periodic orbit, provided T is large enough. Dealing with second order systems of the form

q"(t) = -VV(q(t)), Felmer and Silva [FS981 have proved similar statements assuming growth conditions on the potential V E C2(R'v) of the kind D2V (q)y . y ? OEUI2,

for lql ? R, y E 1',

(4.19)

where Y is a one-dimensional subspace of RN. Assumption (4.19) is weaker than (h3') first because it involves no lower estimate on the directions other

than Y, and second because it is a rank-one condition, and not a ranktwo condition. The first difference is related to the fact that the second order operator d& /dt2 is non-negative, so that solutions of the above system always have non-negative Maslov index. The rank-two condition is required in the first order case essentially because a symmetric matrix B with rank one always gives a linear Hamiltonian system w' = JBw with Maslov index

-N, no matter how large the norm of B is.

4.2.1

Minimality of the period

Neither Rabinowitz's original statement nor our Theorem 4.2.2 imply the existence of a solution of minimal period T, for every T > 0. In principle, system (4.10) could have solutions of period T1, for every Tl belonging

4.2. Superquadratic Hamiltonian

109

to some interval ]0. To[. so that it also has T-periodic solutions for any T > 0, because T = kTt for some k E N and some Tl EJO. To[. However. Rabinowitz conjectured that, under assumptions (hl), (h2) and (h3), the Hamiltonian system (4.10) has a solution of minimal period T for every T > 0 [Rab78a].

This conjecture is still open, but a positive answer was given under more conditions on the Hamiltonian. The first result in this direction is due to Ambrosetti and Mancini [x1181]. A more general statement was proven by Ekeland and Hofer [EH85], who showed that if the Hamiltonian satisfies (hl). (h2). (h3). and is strictly convex. the conjecture holds true.

Their method was to use the dual action principle and the topological characterization of mountain pass critical points found by Hofer [Hof85]. A technical problem arising from this approach comes from the fact that the dual functional is not twice differentiable.

The proof we present here is due to Dong and Long [DL97]. The direct action functional is used, together with the Morse index estimates of Theorem 4.1.9 and iteration inequalities on the Maslov index.

Theorem 4.2.3 (Ekeland. Hofer [EH85)) Assume that H satisfies either (hi). (h2). (h3) or (hl'), (h2'), (h3'). Furthermore, assume that D2H(z) is strictly positive for every - 0 0. Then system (4.10) has a solution of minimal period T. for every T > 0. Lets be the non-constant T-periodic solution provided by Proof. Theorem 4.2.2. or by Theorem 6.10 in [Rab86]. Then /.LT(z) < N + 1.

(4.20)

Actually. the estimate for the Maslov index is not provided by Rabinowitz's statement, but it can be easily deduced by using Theorem 4.1.9 instead of

Theorem 4.1.8 in the proof. Assume that i has minimal period T/k. We will prove that k = 1. Since H is strictly convex and autonomous,

iiI(z) > N.

vi (5) > 1.

(4.21)

the first estimate following from Proposition 3.2.3. By (4.20) and (4.21). Proposition 1.4.4 implies that k = 1. 0

We wish to emphasize that the above proof uses just the fact that the Maslov index of the solutions satisfies the inequalities (4.20) and (4.21). Therefore. any T-periodic solution of an arbitrary Hamiltonian system whose Maslov index satisfies the same inequalities would have minimal period T (see also [LLOO]). For example, in order to have (4.21). it is enough to assume that D2H(z(t)) is positive for every t. and strictly positive for some t. regardless on the behavior of H far from z(t).


110

4.3 A Birkhoff-Lewis type theorem Consider now a time-dependent periodic Hamiltonian H E C2(S1 x R2N), where Sl = R/Z, and assume that 0 is an equilibrium point for the system

z' = Ja-H(t, z),

(4.22)

meaning that H(t, 0) = 0 for any t. By adding a function of t, which will not modify the Hamiltonian system, we can write H as

H(t, z) = ZB(t)z z + H(t, z), = o(1z12). The linwhere B is a loop of symmetric matrices and HO (t, earization of (4.22) at 0 determines a path -y in the symplectic group that satisfies

y'(t) = JB(t)y(t). 1 y(0) = I.

(4.23)

It is a classical problem to determine conditions on -y and on the nonlinearity H which ensure the existence of infinitely many periodic orbits, with larger and larger periods, in arbitrarily small neighborhoods of 0. A classical answer to this question is provided by the Birkhoff-Lewis theorem (BL33], whose statement we are now going to describe, following Moser's approach [Mos77J.

Let 4 : R21V -r R2N be the time-one diffeomorphism determined by system (4.22): +(zo) = z(1), where z is the solution of (4.22) such that z(0) = zo. We will assume that H is of class C'. so that 4' is C3. The point 0 is a fixed point for 0 and d4;(0) = 7(1). Assume that y(1) has at least a pair of eigenvalues o f modulus 1 and that, denoting all the eigenvalues of modulus 1 (repeated according to their multiplicity) as A1.... , A,,,, Ai 1 , ... ,Am the following non-resonance condition holds m

II A7' ,-

m 1

if n j are integers such that 1 < E Inj 1 j=1

j=1

- 4.

(4.24)

In particular, these eigemvalues must be simple. It can be shown that, if (4.24) holds, there exists a symplectic change of coordinates such that 4' can be written in the so called Birkhoff normal form Z = (p q)

N ®(cos¢j(z) -sin i(z) 1 (P

sin Oj(2)

cos Oj(z)

J

qj I

4.3. A Birkho$-Leis type theorem

111

where 4' vanishes up to its third derivatives and

Pi(P.4) = aj + E 33k(pk +9k k-l

for suitable numbers aj and 32k. Then Moser's version of the BirkhoffLewis theorem is the following (see also ([Har68], [Zeh87] and section 1.8 in [HZ94]).

Theorem 4.3.1 (Moser [Mos7 7 ]) Assume that the non-resonance condition (4.24) holds and that the matrix (3;k). appearing in the Birkhoff normal form of I. is invertible. Then in any neighborhood of 0 there exist infinitely many periodic orbits of 4'. It is somehow remarkable that variational methods, which one would probably consider unsuitable to deal with a perturbative problem as this one, provide a result of this kind: the regularity assumptions and the conditions on the linear part are actually weaker, although the assumption on the nonlinearity is quite strong.

Theorem 4.3.2 (Benci, Fortunato [BF87]) Let H E C2(R/Z;R21') be a Hamiltonian of the form

H(t, z) = 2B(t)z z + H(t, z), where B(t) is a loop of symmetric matrices and H(t, z) = OQZ 12). Let -y be the solution of (4.23) and assume that

(i) every symplectic matrix sufficiently close to 7(1) has at least an eigenvalue of modulus 1;

(ii) there exist constants r > 0. c > 0 and 9 > 2 such that >c{zle,

if jzj < r, for any t. Then, for n large enough, the system (4.22) has an n-periodic solution z $ 0 and z,, -+ 0 in C' (R; R2N). If, moreover, I is not an eigenvalue of ry(1), then for any large prime number n, the minimal period among the integers is n.


112

A few comments on the assumptions. before proceeding with the proof. In an earlier paper. Rabinowitz showed the existence of infinitely many periodic orbits in any neighborhood of zero. when B - 0 [Rab80]. Assumption (i) is fulfilled when 7(1) has an eigenvalue of modulus one whose Krein-signature has not the form (p. p), as it can be seen from the discussion of section 1.3.1 (see also [Eke9O] Theorem 1.2.10). In particular. (i) holds when y(1) has a simple eigenvaiue on the unit circle. so (1) is strictly weaker than (4.24), which requires all the eigenvalues on the unit circle to be simple. Having at least one eigenvalue of modulus one is clearly necessary: the time dependent Hamiltonian on R2

H(t.p.q) =

2p2

- 2q` +f(t)(PI +qt).

with f one-periodic and inf f > 0. satisfies all the other assumptions but the corresponding system has no n-periodic solutions close to 0. Indeed, if (p, q) is an n-periodic solution of

pq-f(t)g3.

1 q' = p + f (t)p3,

multiplying the first equation by q. the second by p and integrating on [0, n], we get

dt = jPQ'dt =

j

/'p'g

q2 dt - 1

f q4 dt,

0

jP2dt+jfP4dt.

A comparison of these identities and an integration by parts yield

jq2dt=jfq4 -1 2dt-1 fp4dt < 11Jr q. 0 0 0

where Al := sup f . So there exists to such that jq(to) > AI". Assumption (ii) is again Rabinowitz's superquadraticity. which is now required to hold for the nonlinear part H. in a neighborhood of 0. It is much stronger than the generic assumption on (.33k) appearing in Birkhoff-Lewis theorem. Of course. the presence of some nonlinear term H is necessary because a linear system may have no periodic solutions at all, apart from 0, even if all its eigenvalues have modulus one (they may not be roots of 1).

Finally, the conclusion on the minimality of the period does not allow us

to exclude that zn may have period n/m. for n/rn I N. This is an unlikely

4.3. A Birkhoff-Lewis type theorem

113

circumstance when H depends on time and 1 is its minimal period, but it may happen quite easily if H is autonomous. In order to prove Theorem 4.3.2, it is useful to introduce the unbounded operator S" on L2 (R/nZ; C2N),

S'z = -Jz' - B(t)z, with domain H'(R/nZ; C2N). As we have seen at the end of section 3.2, S" is a self-adjoint closed operator with discrete spectrum, consisting of eigenvalues of finite multiplicity.

Lemma 4.3.3 If assumption (i) of Theorem 4.3.2 holds, then, for every n large enough, S" has an eigenvalue a such that it/n < a < 37r/n. Proof.

Let y be the solution of the linear Cauchy problem

J Y'( = J[B(t) + ;; I]'y,,, (0) = I.

Since y(1)

assumption (i) guarantees that, for it large enough, y, (1) has an eigenvalue of the form e'w, w E R. Fixing such a large n, we deduce the existence of a solution v : R -+ C2N of the system

-Jv'-B(t)v-

21rv=0, n.

such that v(1) = e'°'v(0),

J0

Iv(t)12 dt = 1.

(4.25)

Let a E [-7r, 7r] be such that e"- = e'° and set u(t) := a-' . 1 v(t).

A direct computation shows that u is an n-periodic solution of

-Jul - B(t)u - n

i-.1u.

(4.26)

71,

Since the operator S" - ;;I is self-adjoint, (4.25) and (4.26) imply

o,,:=inflcr(S't-2iI)I= (S _ it

27rI)u

1 IIullLz(O.n)

R such that

aaH(t, z) - Boo(t)zl = o(Izl) for'zJ-i oo, where B. is a loop of symmetric matrices. In this chapter, we will study existence and multiplicity results for periodic solutions of such systems. We will find periodic solutions by using Morse theory to study the critical points of the action functional eH. The main difficulty is that such functional is strongly indefinite, so Morse theory cannot be applied directly: we have seen that the Morse relations do not detect critical points with infinite Morse index. Several different approaches have been developed in order to overcome this difficulty. Conley and Zehnder [CZ84J used a sort of Lyapunov-Schmidt method called saddle point reduction, introduced by Amann [Ama791. A drawback of this approach is that one has to assume

D2H to be bounded. Other methods, based on generalized cohomology theories or on a Morse homology approach, are briefly described in section 5.4.

Here we will use what is probably the simplest possible approach: a Galerkin method, together with the (PS)' condition. The Hessian of H is not required to be bounded, but just to have polynomial growth, so that the action functional is C2. Moreover, this approach fits particularly well with the concepts of relative dimension and relative index developed in the second chapter, the connection being provided by the results of section 2.5. 119

Chapter 5. Asymptotically linear systems

120

Non-resonant systems

5.1

A one-periodic Hamiltonian system

z'(t) =

z(t))

(5.1)

is asymptotically linear if

(AL) there exists a one-periodic loop of symmetric matrices B,, such that

1d

o IzI

a H(t,z) - B,,(t)zl = 0,

uniformly with respect to 1.

The linear system JB,,(t)uw(t)

(5.2)

is called the linearized system at infinity, and for such a system we can consider all the concepts introduced in the first chapter.

Definition 5.1.1 The asymptotically linear Hamiltonian system (5.1) is degenerate at infinity if the linear system (5.2) is degenerate, it is nondegenerate at infinity in the opposite case. The nullity at infinity is the nullity of system (5.2) and it is denoted by v(ac) := v(Bx). The Maslov index at infinity is the Maslov index of system (5.2) and it is denoted by p(oo) := p(B,). The mean winding number at infinity is the mean winding number of system (5.2) and it is denoted by r(oc) := r(B,). Systems which are degenerate at infinity will be also called resonant at infinity. Systems which are resonant at infinity are more difficult to study because the corresponding action functional may not satisfy (PS): indeed. if the system is linear and degenerate, it has a linear subspace of periodic

solutions, so the action functional ell has a non-compact continuum of critical points in eHh({0}). Our first aim will be to prove the Morse relations

for systems which are non-resonant at infinity, an important result due to Conley and Zehnder [CZS4J. We begin with some abstract results about asymptotically quadratic functionals.

5.1.1

Abstract asymptotically quadratic functionals

Let E be a separable Hilbert space and consider a functional of the form

f(z) = 2 (S,, z, x) + h(z), where

.r E E,

5.1. Non-resonant s3-sterns

121

(f 1) Sx is an invertible self-adjoins operator:

(f2) h E C2 (E) has compact gradient: (f3) IIVh(.c)II = o(IIzII) for I)rII -+ x.

By (f 1) and (f2). f has the form analyzed in sections 2.4 and 4.1.1. Furthermore. (f3) provides us with some information on the behavior of f at infinity. Denote by Ez and Ex the positive and the negative eigenspaces of S. We recall that a Lament polynomial is an object of the form ,,,

Q(A) = E aqX . where n. m E N. Here is the main result of this section.

Theorem 5.1.1 Assume that f satisfies (f 1). (f 2) and (f3). If the critical points off are non-degenerate, then Crit(f) is a finite set and the following Morse relations hold: ,rECrit (f )

where Q is a Laurent polynomial with non-negative integer coefficients.

be an approximation scheme with respect to S. in the sense Let of Definition 2.5.1. and set E := R(P ).

Lemma 5.1.2 Assume that (f 1). (f2) and (f3) hold. Then f satisfies (PS)' on R. with respect to Proof.

Let

C E be such that r E E and P, f

0. By

(f3) there exists e > 0 such that IIVh(z)II

2115Xt II 11XII

+ C.

So. for n large we get 1 > IIPTICf (XI)II = IISXr + [P.. SC ]x,. + I t 211Sx 11

IIXnII-II[J'..S JIIIlz,II-r.

Since [P1,.S,) tends to zero. the above inequality implies that (r,) is bounded. Since f consists of the sum of a non-degenerate quadratic form


122

and of a term with compact gradient, (xn) must be compact (see section 4.1.1).

The (PS)* condition ensures that every sequence of critical points of the restricted functionals f,, := f I E converges, up to a subsequence, to some critical point of f. We shall need also a converse statement: every critical point of f can be found as the limit of a sequence of critical points of A. Lemma 5.1.3 Assume that (f 1) and (f 2) hold. Let;:- be a non-degenerate critical point of f. Then there exists no E N and r > 0 such that, for every n > no, the set En fl Br(t) contains a unique critical point xn of fn. Furthermore, xn is a non-degenerate critical point of fn, xn -+ x and b'n > no.

ntE-(x; f) = m(xn;.fn) - 2(PnSSPi,),

The idea of the proof is simple: all we have to do is to use the implicit function theorem to prove that the set {x E E I Pn0 f (x) = 0} intersects En transversally, near x. The unique point of intersection will be the required constrained critical point xn. Proof. By Proposition 2.5.3, there exist n1 E N, S > 0, c > 0 and

ro > 0 such that, if n > n1 and x E Bra(t), Vu E E,

(5.3)

iE- (D2f (x)) = i(P.D2f (x)Pn) - i(PnS.pn),

(5.4)

IID2f(x)II < c.

(5.5)

IIPnD2f(x)PnuII > SIIPnuII,

Let n > n j and set

r'n :_ {x E Bro(x) I PnVf(x) = 0}, so that

Crit(fn) n Bro(x) = r fl En.

5.6)

Then It E Fn and rn is the set of zeroes of the C1 map

4in:EDxHPnVf(x)EEn. The differential of 4;n in x E E, restricted to En, is PnD2 f (x)Pn. By (5.3), (5.5) and by the implicit function theorem, the connected component of r'n containing .t is the graph of some map 91n E C1(Iln; En),

nn C E,i, ,

such that

12n((I -

Pnx,

Iid'Pn(y)Iic(E.

a,

Vy E SZn.

(5.7)

5.1. Non-resonant systems

123

By (5.7), setting rl :_ bra f VrcT_+_F, we may assume that

E n Br, ((I - P.)2) C On. Consider the sets

E.

rn \ {y + wn(y) I Y E E;, n Br1((I - Pn)x)} .

Since En+i C En, we can find r EJO,r1J such that lix -ill > r for every x E En and every n > n1. Therefore, for every n > nl,

rn n Br(ti) = {y + $n(U) I Y E E n Br((I - Pn)z)} n Br(x).

(5.8)

Let no > ns be so large that dis t (2 . En )

< _ + az,

Vn

> no.

( 5 .9 )

Let n > no. We claim that

FnnBr(2)nEn ={xn}.

(5.10)

Indeed, by (5.7),

Il4n(0) - 2112 = II(I - Pn)2112 + IIpn(0) - 'kn((I - PP)x)II2

< (1 +

a } II(I - P),II2,

(5.11)

so, by (5.9).

2. Indeed, (i), (ii) and (iii) are fulfilled when n avoids a finite set of proper ideals, while (iv) holds for n large enough, by Theorem 1.4.3. Since the system is two-dimensional. Proposition 1.4.5 implies that the Maslov index at time it of a one-periodic solution z can be expressed as p,,(z) = [[nr(z)]], for every n such that z is non-degenerate at time n ([[0]] equals 9 if 9 E Z. the closest odd integer if 9 V Z). By (i), (ii) and (iii), the Morse relations for system (5.1) at time n are E A[(nr(z)J} _

(1

A)Qn(A)

_EV

where P denotes the set of one-periodic solutions. Since #P > 2. the right-hand side contains two consecutive monomials Aq, A91'. while on the left-hand side, by (iv), the degrees of two different monomials must differ 0 by two or more. This contradiction proves the theorem. Motivated by Franks' theorem, we think that the above conclusion holds true also without assuming the one-periodic solutions to be non-degenerate. Furthermore, we believe that, at least under some non-degenerance assumptions, the above result should be true in any dimension.

5.2. Morse relations for autonomous systems

5.2

131

Morse relations for autonomous systems

Consider the autonomous asymptotically linear system

z'(t) = JVH(z(t)).

(5.14)

Since system (5.14) is autonomous, the condition of asymptotical linearity becomes:

(AAL) there exists a symmetric matrix B such that

IVH(z) - B.z( = oflzI),

for IzI - +oo.

Critical points of H are equilibrium solutions of system (5.14), hence they are T-periodic solutions for every T > 0. Denote by Do the set of equilibrium solutions of (5.14) and by PT the set of non-constant T-periodic solutions. As usual, T may not be the minimal period of z E PT.

Let zo E R2:v be a critical point of H and let fali, ... , fahi be the purely imaginary eigenvalues of JD2H(zo). Then zo is non-degenerate at time T if and only if o T 27rZ for every j = 1,.. -, h (see Theorem 1.5.1). If D2H(zo) is not invertible, zo is always a degenerate solution. If D2H(zo) is invertible, then zo is non-degenerate at time T for every positive T, apart from a countable set. Since in this section we will study systems which are non-resonant at infinity. we will assume that B,,, is invertible. On the contrary, a non-constant T-periodic solution z is always degenerate: in fact z' is a non-trivial T-periodic solution of the linearized system at z. Therefore. we give the following definition.

Definition 5.2.1 A non-constant T-periodic solution z of an autonomous Hamiltonian system is non-degenerate at time T in the autonomous sense

if v-r(z)=1. Every T-periodic solution z which is non-degenerate in the autonomous sense, generates a critical manifold

{z( +s)18E[O,Tj}, which is diffeomorphic to S' and is non-degenerate in the sense of Definition 4.1.4. Here are the Morse relations for an asymptotically linear autonomous Hamiltonian system.

Theorem 5.2.1 Assume that H E C2(R2''1) satisfies (AAL) and (PG). Assume that system (5.14) is non-resonant at infinity at time T. If all the constant solutions are non-degenerate at time T and all the non-constant T-periodic solution are non-degenerate at time T in the autonomous sense,


132

there exists a Laurent polynomial Q with non-negative integer coefficients such that the following identity holds: )IIT(Z0) -oEPo

+

(1 + A)Q\^)'

(1 ZEPT

The reader may prove the above theorem, arguing as in the proof of Theorems 5.1.1 and 5.1.5 and using the Morse relations for functionals whose critical set consists of non-degenerate critical manifolds (Theorem 4.1.6). The term (1 + \) multiplying \,UT(Z) comes from the fact that the Poincare polynomial of S' is 1 + A. A proof obtained by using a generalized cohomology approach can be found in [Abb00].

5.2.1

Growth of the number of periodic solutions of autonomous systems

Let zo be a critical point of H. We will consider oo as a (virtual) critical point, setting D2H(oo) = B. If zo E R2N U {oo} is a non-degenerate critical point of H, let (p(zo), q(zo)) be the signature of D2H(zo): p(zo) is the number of positive eigenvalues of D2H(zo), while q(zo) is the number of negative eigenvalues of D2H(zo). Since the dimension of the space is even, p(zo) - q(zo) is always even.

Definition 5.2.2 A non-degenerate true critical point zo E R2N of H is called positive if the integer I(p(zo) - q(zo))

is even. Otherwise it is called negative. called positive if the integer 1

2

The virtual critical point oo is

(p(oo) - q(oo))

is odd. Otherwise it is called negative.

Theorem 1.5.1 and Proposition 1.5.2 imply that a true non-degenerate

critical point zo is positive if and only if its Maslov index at time T is even, for every T such that zo is a non-degenerate solution. The opposite characterization holds for the virtual critical point oo. Proposition 5.2.2 Assume that (AAL) holds and that all the critical points of H (true or virtual) are non-degenerate. Then the number of positive critical points equals the number of negative ones (true or virtual). In particular, there are 2k - 1 true critical points, k - 1 of which are positive, k - 1 negative and one whose sign is the opposite of the sign of oo.


133

Proof. By (AAL), H is an asymptotically quadratic function, in the sense of section 5.1.1, so its Morse relations are

r '\q(zo)

= \q(-) + (1 + )\)Qo(A)

zoECrit(H)

Multiplying each member of the above equation by A-N and evaluating it for A = -1 we get

E zoECrit(H)

(-1)q(zo)-N - (-1)q(oo)-N = 0.

The thesis follows from the fact that zo is positive if and only if q(zo) - N is even, while oo is positive if and only if q(oo) - N is odd.

Let x1, ... , xk be the positive critical points of H and let y1, ... , Ilk be the negative ones (oo is included). Assume that they are ordered in such a way that their mean winding number is non-decreasing:

T(xl) < T(x2) < ... < T(xk),

T(y1) < T(y2) < ... < T(yk).

(5.15)

Definition 5.2.3 The global twist of system (5.14) is the non-negative number j=1

When the global twist is strictly positive, it provides us with an estimate on the growth of the number of T-periodic solutions. Next theorem generalizes a similar statement for second order Lagrangian systems [BF97].

Theorem 5.2.3 (Abbondandolo [AbbOO]) Assume that H E C2(R2N) satisfies (AAL) and (PG). Then there exists C > 0 such that, if the system is non-resonant at time T, all the constant solutions are non-degenerate at time T and all the non-constant T-periodic solutions are non-degenerate at time T in the autonomous sense, then the following estimate holds

#PT>

2©T-C.

This theorem can be easily deduced as a corollary of the following more precise result.

Theorem 5.2.4 Assume that H E C2(R21) satisfies (AAL) and (PG). Let T > 0 be such that the system is non-resonant at time T, all the constant solutions are non-degenerate at time T and all the non-constant Tperiodic solution are non-degenerate at time T in the autonomous sense.


134

Let x 1, ... , Xk and yl,... , yk be the positive and negative critical points of H, including oo, ordered so that

pT(yl) < ... < IiT(yk)

µT(21) < ... < IIT(xk), Then

-k2_1

k

#PT J=1

When there is only one equilibrium point, the assumption on the nonconstant solutions being non-degenerate can be dropped, as it has been shown by Izydorek in [IzyOOb]. See Theorem 7.2 in (MW89] for the case of a convex Hamiltonian. Proof. Set

E AYT(O.

W(A)

ZEVT

We must estimate #PT = W(1). Assume that the virtual critical point oo is negative, so that oo = y, for some s E { 1, ... , k}. The other case can be treated similarly. The Morse relations for system (5.14) can be written as k

[,\PT(z,) + \P7(VJ)J + avrfs.) - 1+T(oo) = (1

j=1 jog Set

X := {7 # s l /uT(xj) > 1JT(y1)},

Y := {1 # s I IpT(xj) < pT(yj)} .

Since,ur(xj) is even and pT(yj) is odd, for j 0 s, X U Y = {1,...,k}\{s}. Set

C(a)

'_

X jEX

PT(--j)-1

'\PT(z,) +'\PT(D))

1+A

=

(-1)i+lai,

E jEX i=PT(Y,)

+\µr(Y,)

PT(11,)-1

jEY i=pT(xj)

jE}"

S(a)

,\Pr(=.) - \PT(=) 1+A

11 si'V

=

iEEG

Here G = {PT(co),...,PT(x,) - 1} and si = (-1)i+l if iT(z,) > IAT(co), 1) and si = (-1)i if IIT(x,) < UT(oc)while G =


135

Clearly S = 0 if pT(Xs) = PT(oo). Set

Q(A) = E qia`, W(a) = E wia' i i Q(A) - W (A) _ E riai = F(qi - wi)A.

R(A)

i

i

The Morse relations can be rewritten as R(A) = C(a) + D(A) + S(A).

(5.16)

Given a Laurent polynomial P(A) _ >i piA1, let P- denote the sum of the absolute values of the negative coefficients of P:

P- := - E pi. pi 0 and qi > 0,

#PT=>wi> > wi= >(qi-ri)>->ri=R-. i

r, IIS-xII - II[Pn, Soo]IIIIzII

l

(01

- II[P6, San]II)

11-T11,

and, since [P, , Sam] -3 0 in norm, we have just to consider a new approximation scheme obtained from the old one by neglecting a finite number of projectors. Let E, and E, be the positive and negative eigenspaces of PPS,,,PT, so

that Fn = En ® E,-, and let P and Pn be the corresponding orthogonal projections. Finally, set fn := f I E.. Although So, is not invertible, assumptions (g2) and (g3) guarantee

that f satisfies the (PS)' condition.

Lemma 5.3.2 If (g1), (g2) and (g3'), respectively (g3+), hold, then f satisfies (PS)' on R U {+oo}, resp. on R U {-oo}, with respect to (Pa). Proof. Assume that (g3-) holds. Let (xn) C E be such that xn E En, PPV f (xn) -, 0 and f (xn) is bounded below. We can write Xn = zn+yn , where z,, E E°x and y,, E Fn. The vector

PnV f (xn) = S.y. + [Pn, Soo]yn + Pn Vh(xn)

(5.19)

is infinitesimal. Thus, for n large, using also the facts that Sa is invertible on F, [Pn, Sam] -+ 0 and Vh is bounded, we get 1 > IIPnVf (xn)II > IISoovnII - II[Pn,Soc]IIIIynII - IICh(xn)II > 6II roll - c,

for suitable constants d > 0 and c > 0. Hence (yn) is bounded. Then, since f (xn) is bounded below, also h(xn) is bounded below and (g3-) implies that (zn) is bounded, hence compact, because E°,. is finite dimensional.

Therefore, (Vh(x,,)) is compact and the fact that (5.19) is infinitesimal implies that also (yn) is compact. An analogous argument proves (PS)' on RU {-oo} when h satisfies (g3+). Proof.

[of Theorem 5.3.1.]

Assume that (g3-) holds, the case with (g3+) being analogous. By (5.18),

(V fn(x),

(PnSwx + PnVh(x), P,s x) 20, IIPn xII2 - IIVh(x)IIIIP,n xII,

5.3. Systems with resonance at infinity

139

hence, since Ch is bounded, there exists R > 0 such that 0

if

R.

(5.20)

From (PS)' we know that f is bounded above on Crit(f ), but we may assume that if is also bounded below, because otherwise f would have infinitely many critical points and there would be nothing else to prove. So let a < b be real numbers such that

Crit(f) C f-'(]a,b[). By (g2) and (g3-), the estimate

f(x)

2IISoIIIIP+xII2

- 2oIIP xIIZ+h(P°z)

+(IIP+zII + IIPTxII) sup IIVhII,

shows that f (x) -+ -oc for II(P° + P- )zII --* oc, uniformly on every set where IIP+xll is bounded. Therefore, there exists Rl > 0 such that

IIP+zII < 2R and II(P°+P-)xII >

2'

f(x) 0 and n(oo) V [mEm (zo), mEW (xo)];

then f has at least another critical point.

The idea of the proof is simple, but some technicalities are needed. Therefore, we just sketch the argument, referring to [CL90J for details.

Condition (g4) guarantees that the (PS)' condition fails only at level zero.


141

Lemma 5.3.4 If (gl), (g2) and (g4) hold, then f satisfies (PS)' on R\ {0}. Furthermore, if (sn) is a non-compact (PS)' sequence such that f (rn) -+ 0. then. up to a subsequence. IIP°rnll -+ ze and (I - P°)r" -> 0. Proof.

Let (r,,) C E be such that r" E E. and PnV f (x") -+ 0.

Write rn = z + yn. where zn E F°x and Y. E F. Arguing as in the proof of Lemma 5.3.2. we find that (yn) is bounded. If (z,,) is also bounded. (y,,) is compact. Otherwise we may assume that IIzn(I -+ oc and (g4), together

with the fact that P,V f (zn) = S. Y, + Pnch(rn)

is infinitesimal, implies that yn -+ 0. Since h(rn) -+ 0. we conclude that

f

0

0.

Sketch of Proof [of Theorem 5.3.3.]

We can compactify E°, by adding the point x and obtaining the manifold

E2

Set 11 := E°x x F and consider the functional g ::If -+ R defined as g(z.y)

f(z+y).

if

>(S.y.y). if

z#x.

z

=x.

The function g is continuous and. although it is not, Ct. the usual defor-

mation lemmas hold. the critical set of g being Crit(f) U {(x.0)}. Set 11,, := E°x x Fn and gn := g1tit . There exist no E N and R > 0 such that. setting

U,,.RE°, }n.R:=Ev, x(BRnEn)x(c7BRnEn). (('n.R.1n.R) plays the role of an index pair for g,, for every it > n°. The' pair (L',1,!(.1n.R) is continuously retractable onto (E°x x (BR n E ). E°x x (e8BR n E. )) ` (S"e x1 x B. Snr -' x OB).

where B is a closed ball of dimension qn := dim E = i (P S,, P,,). By the Kunneth formula, the homology classes spanning H°(S' I) and H,,( )(S"' generate two non trivial classes an E Hq,,(Z'n,R,1n.R)

and

Q,*, E

Hq.tntx)((n,R1n,R)


142

Then

c, := inf supgn and c,, := inf sup9" fEa. IEI

CEO; Ifl

are critical levels of g,,, such that c, < c;,. Since R is independent on n, (c,,) and (c,,) are bounded and we may assume that they converge to the same values c < c'. If c = c' = 0, it can be proven that g, and thus f, has infinitely many critical points. This is due to the fact that the homology classes an and a,, are subordinate, meaning that there exists a cohomology class w, E H"(0D)(Mn) such that a;, fl w,, = an (here it denotes the cap product). In this case, either the strict inequality c" < c,, holds or gn has infinitely many critical points (see Corollary 11.3.3 in [Cha931). This dichotomy holds also after taking a limit on n, because the difference c;, - c" can be explicitly estimated (see Corollary 2.5 in [CL90) and Corollary 11.3.2 in [Cha93j).

So we can assume that at least one of c and c' is different from zero, hence g has a critical point other from (oo, 0), which must be a critical point of f. This proves the first statement and, since the level of such a critical point is not zero, also the second statement under assumption (i). To deal with (ii) and (iii), notice that at least one of these inequalities must be true: c < 0 or c' > 0. Assume that c < 0 holds, the other case being analogous. The restricted functional g,, has a critical point xn such that g,, (x,,) = c,, and

m(xn;9n) < qn : m'(xn;9n). Since g satisfies (PS)', up to a subsequence (x,,) converges to a critical point 2 of g such that g(a) = c. Moreover, Theorem 2.5.4 and the semicontinuity properties of the Morse index imply that ME;. (x; g) : 0 < ME- (.t; g).

Since g(2) < 0, x E E, and it is a critical point of f such that f The above estimate on its Morse index shows that 2 # xo.

0. O

Our last result requires no assumptions on the behavior of h and 6h, but only on D2h: (g5) D2h(x) -3 0 for IIP°xII -- oo, uniformly on sets where II(1- P°)xII is bounded. The above assumption is too weak to guarantee the existence of one critical

point: f(z) = arctanx satisfies (g1), (g2) and (g5), but has no critical points, However, (g5) allows us to prove the existence of a critical point once the presence of another one, satisfying suitable Morse index estimates, is known.


143

Theorem 5.3.5 (Abbondandolo [Abb00]) Assume that f satisfies (g2) and (g5). If f has a non-degenerate critical point xo such that

(gI),

mE.- (xo; f) 0 [-1,n(oc) + I], then f has at least another critical point. Proof.

Assumption (g5) does not allow to localize those levels where

(PS)' fails, therefore, we will modify f so that (PS)' will hold. Let cp E C' (R) be a non-decreasing function such that cp(s) = 0 for s < 0 and

W(s) =s fors> 1. For R>Oset fR (x) = f(x) ± '(II1'0x112 - R2).

Claim 1. The functional fR satisfies (PS) on A. Let (xn) C E be such that C f R (xn) -+ 0. Write xn = an + yn, where zn E E°,,. and yn E F. Then V fR (xn) = Sacyn + Vhi(xn) ± 2jp'(IIznII2 - R2)zn.

(5.22)

Projecting onto E°., we get that

P°Vh(xn) ±29'(Ilznll2 - R2)zn is infinitesimal, so

must be bounded. The inequality

IIVfR (xn)II >- IlSxynll - IIVh(xn)Il - 2supIcp'IIIznII

implies that also (yn) is bounded. Therefore, (zn) and Vh(xn) are compact and the fact that (5.22) is infinitesimal implies that (yn) is compact. Actually, fR satisfies also (PS)' on R, but for now we do not need this fact. We will need it later, but for a slightly different functional.

Claim 2. There exists Q > 0 such that. if IRI - P°)xII < c := 2v sup IIVhII

and

IIP°xll ? Q

then

i

(D2f (x)) < n(om),

for every R > 0.

Indeed

D2 fR (x) = Sw + D2h(x) + K(x),

(5.23)


144

where K(x) is a self-adjoint operator whose rank is E.. Choose Q > 0 so large that

II(I - P°)xII : c, IIPoxII ? Q

h(x)11:5

I 2Q

Then D2 fR (x) is strictly positive on E. and strictly negative on

so

(5.23) follows.

Fix Q as in the above statement: we may assume that IIP°xoll < Q. Consider the functionals fQ. Assume, by contradiction, that f has no critical points apart from xo. The functionals fQ and ff coincide with f in the region defined by IIPoxII < Q. so they also have no critical points there, apart from xo. Since fQ satisfies (PS) on R, Crit(fQ) is compact, so there exists e > 0 such that fQ has no critical points in the region Q

IIPOXII < Q + e.

We would like to apply Morse theory to Q, proving the existence of a critical points # xo, whose Morse index does not fit with the estimates (5.23). The existence of such a critical point would give a contradiction. However, we need first to perturb fQ in order to guarantee that it has only non-degenerate critical points, in the spirit of a well-known result by Marino and Prodi [MP75]. Let w E C' (R) be a non-decreasing function such that v/ (s) = 0 for a < Q2 and w(s) = I for s > (Q + e)2. Set w(x)

L(IIP'xII2)

Notice that Vw is bounded. For w E E set 9w (x) := fQ(x) - w(x)(x.w).

Then its gradient is vg"" (x) = V fQ (x) - (x, w)Vw(x) - w(x)w.

Claim 3. If IIwII is small enough, every critical point x of g* satisfies 11(1- P°)xII < c, where c is the constant defined in Claim 2. Indeed, if

(x) = 0,

0 = II(I - P°)Vg'(x)II = IIS.(1- Po)x + (I - P°)Vh(x) -w(x)(1 - P°)wJI > a 11 (I - P°)xtI - sup IIVhII - IIwII,


145

which implies that II(I - P°)xII < c, provided IIwII is small.

Claim 4. If IIwII is small enough, g has no critical points x such that

Q5 IIP°xII < Q + E.

Indeed, this is true for

fG}.

Since fR satisfies (PS), its gradient is

bounded away from zero in the region Q < IIP°xII < Q+e. Then the claim follows from the fact that Vg± -> G fQ uniformly, as w - 0. Ify

Claim 5. If IIwjI is small enough, gLL satisfies (PS)' on R with respect to (Pn).

Let (xn) C E be such that x,, E E and P,,Vgu (x,,) -* 0. Write x = zn + yn, where Zn E EO, and yn E F. Then PnVgw(xn) = PnScx:yn +P,, Vh(xn) ±2cp'(IIz,,II2

- Q2)zn

-(xn, w)Cw(zn) -

(5.24)

Since P°Vgw (xn) -1 0, the sequence P°V h(xn) ±p ' ( I Izn 1 1 2

- Q2 )zn - ( X " , w)V w(zn) - w(zn) P°W

is infinitesimal. the properties of cp and w force (zn) to be bounded. Moreover, for it large, 1 > IIPnVgu:(xn)II > IISxynll - II[Pn,S-]IIIIynIP IIVwII.IIwII(II'!/,.II + IIznll) - II'u'II

The above estimate implies that, if IIwIT is small enough, (y,) is bounded.

Therefore, (zn) and Vh(xn) are compact, and the fact that (5.24) is infinitesimal implies that (yn) is compact.

Claim 6. There exists R > such that, if IIwII is small enough, the pair

((BR-nEn (D EO) x (BRnE72),(BRnE; (@ E2) x (OBRnE,)) is an index pair for gu I E., while

x (BRn(E,

x (OBRnEn Ei}E ))

is an index pair for gw I E The argument is similar to the one used in the proof of Theorem 5.1.1.

Claim 7. Assuming, by contradiction, that f has only the critical point

x°i there exist vectors w E E with arbitrary small norm such that the functionals g+ and gu, have only non-degenerate critical points.


146

The critical point xo is non-degenerate by assumption. Then Claim 4 implies that, if IIwII is small, all the other critical points x of gw satisfy IIP°xII > Q+e. The function w is constantly equal to one in a neighborhood of such a critical point, so V gw (x) = V fQ (x) - w and D2gw (x) = D2 fQ (x).

(5.25)

Since V fQ and V fQ are C' Fredholm maps of index zero, Sard-Smale Theorem implies that the set of their critical values has first category [Sma65a]. Therefore, we can find vectors w with arbitrary small norm which are regular values for V fQ and V f . Then (5.25) implies that, for such vectors, all the critical points of gU+, and gw are non-degenerate.

Conclusion. Choose w so small that Claims 3, 4, 5, and 6 hold. By Claim 7, we may also assume that g,+ and gw have only non-degenerate critical points. Using Lemma 5.1.3 and arguing as in the proof of Theorem 5.1.1, we get that, if n is large enough, gL+, I E. and 9w IE have only nondegenerate critical points which correspond to the critical points of g,+ and 9w through the bijections 3n : Crit(gw) -+ Moreover, for every x E Crit(gw), mEa, (x; 9w) = m(jn (x); 9w 1E.) - i(PPS.PP.).

So Claim 6 allows us to write the Morse relations for ,\mE,-(xo;f)

and gw as

+ E AmE_(x;9u) = 1 + (1 + A)Q+(A),

(5.26)

xEK+

Amr-(xo;f) + E AME_(x;9w)

+ (1 +)t)Q_(,\),

(5.27)

xEEKK-

where K+ and K- are the set of critical points of g,+,, and gw in the region where IIPoxII > Q + e. Claim 2 implies that

mE- (x; 9w) ? 0 Vx E K+, mE+ (x; 9w)

n(oo)

Vx E K-.

If ME- (xo; f) < -1, (5.26) implies that g,+ has a critical point whose Morse index equals ME;, (xo; f) f 1. Since such an integer is strictly negative, this critical point cannot belong to K+ and we find a contradiction. Similarly, if mE_ (xo; f) > n(oo) + 1, (5.27) implies that gw has a critical point whose 0 Morse index equals ME- (xo; f) f 1 > n(oo), another contradiction.


5.3.2

147

Resonant systems

Now we can apply the abstract theorems of the previous section to asymptotically linear Hamiltonian systems which are resonant at infinity. If the Hamiltonian H satisfies (AL), set G(t, z) := H(t, z) - 1 B... (t)z z.

Let E°,. C E = H' (S'; R2N) be the space of one-periodic solutions of the linearized system at infinity

w'(t) = JB.(t)w(t). Denote by P° the orthogonal projection onto E.O. We begin with an easy lemma.

Lemma 5.3.6 There exists 5 > 0 such that inf Iw(t)I ? JIIwII Vw E E.O.

tES'

Proof. Every function w E E°,. is continuous and, by the uniqueness property of linear Cauchy problems, it cannot vanish, unless w - 0. Since E° is finite dimensional, there exists 5, > 0 such that inf Iw(t)I >- 81 sup Iw(t)I

bw E

fES

The thesis follows from the fact that the norms II . II and II' Iloo are equivalent

on E°.. Here is the statement about Landesman-Lazer type conditions.

Theorem 5.3.7 (Chang, Liu, Liu [CLL971) Assume that H E C2(S' X R2N) satisfies (AL) and (PG). Moreover, assume that VG is bounded and that G(t, z) -; +oo, resp. G(t, z) - -oo, for I zI - oo, uniformly in t. Then system (5.14) has at least a one-periodic solution. If (5.14) has a one-periodic solution zo such that A(oo) V [FZ(zo), Ft(zo) + v(zo)), resp. p(oo) + v(oo) V [p(zo), p(zo) + v(zo)],

then it has at least another one-periodic solution.


148

Proof. Assume that G(t, z) -+ +oo, the argument for the other condition being analogous. We can write the action functional as

eH(") =

1(S.z, z) + h(z),

where S,o is the self-adjoint realization of the quadratic form

Jz'(t) z(t) dt -

- J0 on E, and

h(z) :_

(

B. (t)z(t) z(t) dt,

fo G (t, z(t)) dt.

The gradient of h is bounded and we have only to show that h satisfies (g3-): the conclusion will follow from Theorem 5.3.1. Since VG is bounded,

there exists c > 0 be such that

IG(t,z1)-G(t,z2)I 0 and p(oo) + v(oo) V [/c(zo), p(zo) + v(zo)];


149

then it has at least another one-periodic solution. Proof.

Keeping the notations of Theorem 5.3.7, we have only to

prove that

f G(t, z(t)) dt

h(z)

0l

satisfies condition (g4): the conclusion will follow from Theorem 5.3.3. Let

z = w + u E E, where w=Pozandu=(I-P°)z. If Ilull <M,we have that also IIuIIL2 < M, so for every c > 0 there exists a measurable set S(u) C S' whose Lebesgue measure does not exceed e and Iu(t)I
Al which is C'-close to the identity are in 1-1 correspondence with the zeroes of the exact 1-form A,,,. i.e. with the critical points of a function f,, : Al -+ R. Hence a Hamiltonian diffeomorphism C' -close to the identity has at least as many fixed points as the minimum number of critical points of a function on Al: such number will be denoted by D(M). The non-degeneracy condition

on the fixed points translates into the fact that fl., is a Morse function. Hence, denoting by ND(M) the minimum number of critical points a Morse function must have, we see that a Hamiltonian diffeomorphism C'-close to

the identity with non-degenerate fixed points must have at least ND(M) many fixed points. The Arnold conjectures require that, as in the case of arbitrary diffeomorphisms, these estimates hold also for Hamiltonian diffeomorphisms which are not close to the identity.

Conjecture 6.1.1 (Arnold Conjecture 1) Let (Al,w) be a compact symplectic manifold. Every Hamiltonian symplectic difeomorphism :' : Af -+ Al with non-degenerate fixed points has at least ND(M) many fixed points.

Conjecture 6.1.2 (Arnold Conjecture 2) Let (M,w) be a compact symplectic manifold. Every Hamiltonian symplectic difeomorphism jp : M -+ M has at least D(M) many fixed points.

156

Chapter 6. The Arnold conjectures for symplectic fixed points

For now, none of these conjectures has been proven in full generality. However, there has been considerable progress.

Let us start with the case M = S2. In this case, w is just an area form on M and a Hamiltonian diffeomorphism p is a diffeomorphism which is homotopic to the identity and area-preserving. The Arnold conjectures require W to have at least two fixed points. Being homotopic to the identity, gp has a fixed point: otherwise it could be used to build a homotopy

between the identity and the antipodal map, which cannot be homotopic because they have opposite degrees. Topologically, this is the only fixed point required: indeed, seeing S2 as the Riemann sphere, the homeomorphism z o-r z + I has only one fixed point. The existence of a second fixed point is a consequence of the fact that V is area-preserving. Indeed, if there were no other fixed points, we could use a stereographic projection to build a homeomorphism of the plane which preserves a finite regular measure and has no fixed points. By the Brouwer translation theorem, for every homeomorphism rp of the plane with no fixed points there exists a nonempty open set U such that vyk(U) are mutually disjoint, for k E Z (see [Fra92b]). This is not possible when ;5 preserves a finite regular measure. Using a strictly two-dimensional argument of this kind, Eliashberg proved the Arnold conjectures for all Riemann surfaces [Eli79]. The first higher dimensional breakthrough is due to Conley and Zehnder, who proved both Arnold conjectures when Al is the 2N-torus, equipped with its standard symplectic form [CZ83]. The first aim of this chapter will be to show Conley and Zehnder's proofs of these results (for a slightly different approach see the book of Hofer and Zelmder [HZ94]). Another proof,

based on generating functions, can be found in the book of McDuff and Salamon [MS95]. Shortly afterwards, Fortune used a similar approach to prove the Arnold conjectures on the complex projective space [For85]. In the late 80's, Floer's ideas opened the way to proving the conjectures for arbitrary compact symplectic manifolds. See section 6.4 for more comments on Floer's theory and on the present state of the Arnold conjectures.

6.1.1

Hamiltonian diffeomorphisms of the two-torus

Let T2N := R2N/Z2N be the 2N-torus, endowed with its standard symplectic structure wo(C *1)

A . 71,

,f =

(01

01

I

e ...

\

01)

.

0

If 1: R2N -> R2N is a periodic map, meaning that +(z + k) = 4(z) for

6.1. The Arnold conjectures

157

every k E Z2N, we set [4'] :=

J

4'(z) dz.

o.i12N

The following characterization of Hamiltonian diffeomorphisms of (T2,,.do) was stated by Arnold in [Arn78], Appendix 9. The proof we present is due to Conley and Zehnder [CZ83].

Theorem 6.1.1 If p is a symplectic diffeomorphism of (T2.wo) then the following statements are equivalent:

(i) p is a Hamiltonian diffeomorphism;

(ii) y, lifts as a map on Rl of the form z - z+$(z), where 4' is periodic and [4'] = 0.

The periodicity of 4' is necessary in order for 4' to represent a map on the torus and to be homologous to the identity, so the relevant statement in (ii) is that the baricenter of + is zero. In particular. y: cannot be a translation. Let (M,. o) be a symplectic manifold and consider the set

t := { w symplectic form on Al I j .' = j wo for all 2 -cycles a on Af 111

111

Denote by Diff (Af) the group of diffeomorphisms of M. The following lemma uses an argument of Moser's [Mos65].

Lemma 6.1.2 Let t H wt E t be a smooth contractible loop in r such that w, = wo for s = 0,1. Then there exists a smooth loop t '- yet E Diff(M) satisfying ',pr wt = wo

and

;Go = SP1 = id.

Proof. Let [0, 11 x R/Z 9 (s, t) H d,,t E r be a smooth homotopy which contracts to wo: wo.t = wo,

wi.t = wt,

w:,o =

By the Hodge decomposition theorem with respect to a given metric on M (see [dR84]), we can write

a

88Ws.t = As.t + hs.t,

(6.1)

where h,,t is a harmonic 2-form. Since w,,t E 1', the integral of $w,,t on every 2-cycle vanishes, so the same holds for h,,t. Since h,,t is harmonic,


158

hs,t = 0. Since wa,o is constant, we can choose Let X,,t be the unique vector field such that

in (6.1) so that

= 0.

wa.t(Xs.t, ") = -G.f.

and let Va,t be the family of diffeomorphisms satisfying 61

ga ;ps,t =

o ps.t,

po.t = id.

Since wa.t is closed, by Cartan identity we get

a

Ja.!)

[dC8.t + d(wa.t ( ,,t,

49

-Ja.t +

- 'Ps,t [

1

[d

.t - (ks.t] = 0.

a tws t = wo,t = wo. From 9.o = 0. we get that Ps.o = id, so pt := V1,t is the desired loop of diffeomorphisms. Hence

Consider the following subgroups of Diff(T2A ): Do := {yp E Diff(T2N) I w is homotopic to id} , {VEDolwo=wo}. Dl

D2 := {V E Di I

lifts on R2N as z r- z +

where [4o] = 0}

.

Lemma 6.1.3 If N = 1, the sets Do, Dl and D2 are connected by smooth arcs.

Proof. It is a non-trivial and strictly two-dimensional fact that any E Do can be connected to the identity by a smooth are of diffeomorphisms, see [EE69].

If p E V1, we can take a smooth arc [0,1] 30 t H Ot E Do such that t)o = id and V)1 = V. Set wt := vtowo, so that wl = wo. Then wt E F. Since T2 is two-dimensional, r is convex, so wt satisfies the assumptions of Lemma 6.1.2, which provides us with the existence of a smooth loop of diffeomorphisms Bt such that 8f wt = wo and 80 = 81 = id. Therefore, SPt = 8t o tpt is a smooth are connecting id to in Dl.

If, moreover, V E D2, ipt lifts to a map z s- z + at + +t(z), where

atER2,ao=al=0and[4t]=0. Setrs:zz-a.,sothat raoV. connects id to ip in D2.

In view of this lemma, Theorem 6.1.1 is an immediate consequence of the following result.


159

Lemma 6.1.4 A smooth are cpt E Dl such that cpo = id is the flow of a time-dependent Hamiltonian vector field on T2N if and only if apt E D2. Proof. Assume that cRt is the flow generated by the Hamiltonian vector field 11. Being homologous to the identity, cpt lifts to a map (pt R2N -> R2N and it is enough to show that (6.2) dd

[o.ll

Since cpt is volume preserving, we get

J

c3t (z) dz =

o,ll 2N

2N

JVH(t, V, (z)) dz = J J 2N VH(t, w) dw = 0, T

proving (6.2).

Let yot E D2 and let :9t : R2N i R2N, cpt(z) = z + Dt(z), be a lift of cpt such that [fit] = 0 and 4Do = 0. Then cpt is a path of symplectic diffeomorphisms of (R2N, wo). Moreover, cpt is the flow generated by the time-dependent vector field

Y

d_ (it

t ° cpt

.

Since cpt wo = wo, we get that Ly wo = 0 and, since dwo = 0, d(iy wo) = 0. Poincarelemma implies that iywo = -dHt for some Ht E COO(R2N). More explicitly, we can choose

Ht (z) :_ - fo wo(Y (sz), z) ds, 1

so Ht depends smoothly on t. Hence 1 I = JOHt is a Hamiltonian vector field. Since }t (z) is periodic in z, ft,(z) = H(t, z) + v(t) . z, where H is periodic in z and v(t) E R2N. Since [fit] = 0,

0=f o,112N

d cpt (z) dz = f Jv(t) dz = Jv(t), dt 0 112N

so v(t) = 0 and Ht(z) = H(t, z) is periodic in z. On the 2N-dimensional torus the implication (i) (ii) holds, as the above lemma shows, but the author does not know whether the converse statement is true. For more details on the group of symplectic diffeonlorphisms of the torus see [MS95].

160

6.1.2


Non-degenerate fixed points on the torus

The aim of this section is to prove the Arnold conjecture about nondegenerate fixed points on the torus T2N. endowed with its standard syxnplectic structure wo. The homology groups of T2N are Hq(T2N; Z) = Z('Q ),

q = 0,..., 2N.

so Morse theory implies that 2N

ND(T2N) > E rank H.(T2N; Z) = 22N Q=0

The function 2N

J (z1, ... , z2N) = E sin 27rzj, t

(z1.... , z2N) E T2N.

j=1

has exactly 22N non-degenerate critical points, so

ND(T2N) = 22N. Theorem 6.1.5 (Conley, Zehnder [CZ83]) Let :p be a Hamiltonian diffeomorphism of (T2N, Sao) with only non-degenerate fixed points. Then ND(T2!V) = 221V.

Choose a time-dependent Hamiltonian H E C'°(R/Z x T2N) such that V is the time-1 flow generated by the Hamiltonian vector field A'H = JVH. There is a 1-1 correspondence between the fixed points of V and the periodic orbits of the Hamiltonian system

z'(t) =

J--H(t,z(t)). 492

z(t) E T2N.

(6.3)

Furthermore, a fixed point of p is non-degenerate if and only if the corresponding periodic orbit is non-degenerate, in the sense of Definition 3.3.1.

Notice that, if 18H/8zI is small, system (6.3) cannot have any noncontractible one-periodic solutions. For this reason, we are going to look for periodic solutions of (6.3) in the space of contractible loops. More precisely, Theorem 6.1.5 will immediately follow from the following Morse relations.


161

Theorem 6.1.6 Let P be the set of contractible one-periodic solutions of (6.3). If all elements of P are non-degenerate, then

At'(') = C: (2N) aq-N + (1 + A)Q(,1), q=O

rEP

q

(6.4)

for some Laurent polynomial Q with non-negative integer coefficients.

i

We start by modifying slightly the functional setting of Chapter 3, in order to deal with Hamiltonian systems on tori. Denote by it : R2N T2N the quotient projection. Every smooth contractible loop on T2N lifts to a closed loop on R2''v. such a lifted loop being unique up to integer translations. So we can identify the set of smooth contractible loops on T2N with the set T2N x

I

rt u E C-(S';R2N) I ( u(t)dt=01, o

by the map (zo, u) '- zo + a o u. As usual, it is useful to replace the space of smooth loops by loops in the Sobolev space H4, obtaining the Hilbert manifold

l ri A:=T2NxF, where F:={uEEI J u(t)dt=OI rr

0

111111

and E := H 4 (S' : R2N ). The contractible one-periodic solutions of (6.3) are exactly the critical points of the smooth functional

eH(z) = - 2

J1 Ju'(t) u(t) dt - J I H(t, zo + rr o u(t)) dt, 0

0

with z = (zo, u) E A. The tangent space of A at every point can be canonically identified with E. If E = E+ cD E- is the splitting introduced in section 3.2,

E+:=RNA, uEEIu(t)=Ee2akrtuk

,

kk>t

E- := RN (B

e2:kltuk

u E E u(t) =

,

k 2N + 1. The Galerkin reduction used in the last section does not seem the best way to deal with a functional which may have degenerate critical points: even if one proves the existence of 2N + 1 critical points for eHIk,,, one would have to guarantee that taking the limit for n -+ oc one still gets 2N + 1 distinct critical points of e.H. Therefore, we use a slightly different method, known as saddle point. reduction. Proof. Keeping the notations of the last section, we claim that the set {D2b(z) I z E Al is a relatively compact subset of the space of compact operators on E. Indeed, considering b as a function on loops in RIN, we see that b is smooth also on L'2(S';RIN). Let DL2b denote the Hessian of b with respect to the L2-inner product. If j : E -+ L2 (.SI ; RIN) is the natural (compact) immersion, we have, for z, v, w E E, d2b(z)[v,'w] = (D2 ,b(i(z))i(v),i(«))L2 = (z*(DL2b(i(z))i(v)), w),

164


is the inner product of E. So D2b(z) = i' o D',b(i(z)) o i. The family {D2,b(i(z)) I z E A constitutes a bounded subset of the space of continuous operators on L . Now, it is a general fact that if we compose a bounded set of continuous operators B with two compact operators K1 and K2, we obtain a relatively compact set K1 BK2: if K1 and K2 have finite rank, K1 BK2 is a bounded subset of a finite dimensional subspace; in general, given e > 0 and approximating K1, K2 with a-close finite rank operators, we get that K1BK2 is in a E-neighborhood of a compact set, where

hence it is relatively compact. The above fact implies that we can find a positive integer n such that, for every z E A, II(I - Pn)D2b(z)II
0. The function H lifts to a smooth U-invariant function K on Sl X S2N+1. We may think that K is defined on the whole S1 x CN+1 in such a way that for every t E S', K(t, -) is homogeneous of degree 2. Then K is smooth on CN+1 \ (0), it is continuously differentiable everywhere, it is non-negative and U-invariant. Hence

a K(t eiez) = eie a K(t z),

(6.8)

K(t , z) Jz.

(6.9)

az

az

0=

K(t , eiez)l e_o

z

'

Let A E R and consider the Hamiltonian system

x' = J (-it z) + 27rAz) ,

.

(6.10)

6.2. The Arnold conjectures on the projective space

167

We claim that the solutions of (6.10) have constant norm. Indeed, since the Hamiltonian giving system (6.10) is U-invariant and since the U action is generated by the Hamiltonian G(z) := IzI2, by Noether theorem G is a constant of the motion. More directly, if z2solves (6.10),

aK(t,z)+27rAzI =0, Ill

by (6.9). Since OK/8z is homogeneous of degree 1, every non-zero solution of (6.10) can be divided by its norm, obtaining a solution on S2N+1 If z solves (6.10), then w(t) := e-2,,Atz(t) solves

w' = J

K(t,w).

(6.11)

If, moreover, w(t) lies on S21V+1, it o w is an integral line for Xy, the Hamiltonian vector field determined by H on CPN. If z is one-periodic, then also it o w is one-periodic.

Let wi and w2 be solutions of (6.11) such that it owl = it o w2. Then there exists a real function 9 such that w2(t) = e'°' w1(t); differentiating this identity and using (6.8), we get that 8 is constant. If (z1, A1) and (z2, A2) are solutions of (6.10) on S2N+1 such that, setting wi(t) := e-2xiAltzl(t), w2(t) := e-2TiA2tz2(t), the projections of w1 and w2 on CPN coincide, we deduce that w2(t) = e'Bwi (t) for some real constant 9, hence z2(0) = eiez1(0),

e-2xiA2Z2(1) =

eiee-2xiA1z1(1)

If z1 and z2 are one-periodic, these identities imply that A2 - Ai is an integer.

We can summarize the above discussion stating that it is enough to prove that (6.10) has N + 1 non-trivial one-periodic solutions

such that U z1, ... , U ZN+1 are all distinct and A1, ... , AN+1 lie within an interval of length less than 1. Let E := Hi (R/Z; R2N+2) and set r1

M:= zEEI / I

1 Iz(t)I2dt=1./0

111

The hypersurface M is smooth and every constrained critical point zo of eKIM satisfies deK(ZO) = 2irAodg(zo),

168


where Ao E R and g(z) := IIztIia. Using again the Hamiltonian G(z) _ z Izl2, the above identity canz be rewritten as deK+2x.\.G(Z0) = 0,

which is equivalent to (6.10), with A = A0. Since eK+2aAoG is homogeneous of degree 2, 0 = deK+2,AoG(z0)[zo] =

2

f 2xAolzo(t)I2dt = xAo. 0

Therefore, it is enough to prove that eK I M has N + 1 critical points, belonging to different U-orbits, whose critical values lie within an interval of length less than a. Set

En :=

z E E I z(t) = E e2akJt zk

,

IkI 3N + 1.

Indeed, (i) and (ii) are fulfilled when k avoids a finite set of proper ideals of Z, while Theorem 1.4.3 shows that (iii) holds for k large enough. We can introduce the Hamiltonian Hk(t, z) := kH(kt, z), so that the points of period k of V correspond to the one-periodic solutions of

z' = J-Hk(t,z).

6.4. Some bibliography and further remarks

171

Arguing as in the proof of Theorem 6.1.6 and keeping the same notations, we see that the action functional eH,, E C°°(A) satisfies (PS)' on R and, if n is large enough, eH,, IA. has an index pair (U,,,Yn) whose Poincar6 polynomial is

P(Un,YY) = \di-F; P(T2N). In particular, Hq (U,,, Yn) is non-trivial for q = dim Fn . Theorem 4.1.4 implies the existence of a critical point zn of ell. IA. such that m(zn; eH. I.R) < dim Fn -< m*(zn; eH.IA,. )

Up to a subsequence, (zn) converges to a critical point i of eH. and, by the results of section 2.5, ME-(z;eH.) < -N -3N,

(6.16)

and (iii) implies that z has mean winding number zero. However, since z is weakly non-degenerate at time k, Theorem 1.4.3 implies that lik(z)1 < N. This contradicts (6.15), proving the theorem.

6.4

0


The Arnold conjectures Several problems arise when one tries to use variational methods to prove the Arnold conjectures for wider classes of symplectic manifolds. A first difficulty is that the action functional is well defined only when w vanishes on ir2(M). If one considers only the manifolds with this property. the major and unsolved problem seems to be how to find a good functional setting. Since the Hi-functions are not continuous, the loops on M of class Hi do not form a Hilbert manifold (one cannot use the exponential map of Al to build charts). In spaces of curves with higher regularity, the (PS) condition fails. Moreover, it is not clear how one can find finite dimensional reductions in this highly nonlinear situation. Floer's breakthrough was to consider the L2-gradient of the action functional. The action functional is not smooth in L2 (its leading part is not even defined on the whole L2), so the L2-gradient does not generate a flow. However,


172

the evolution equation defined by it is a nice elliptic equation and one can study a suitable boundary problem for it: this problem turns out to be strictly related to Gromov's theory of J-holomorphic curves, see [Gro85]. The numbers ND(M) and D(M) can be estimated below by SB(M),

the sum of Betti numbers of M, and by CL(M) + 1, one plus the cuplength of M, respectively. So the Arnold conjectures can be stated in a weaker form by asking that #Fix(cp) should not be smaller than SB(M), in the non-degenerate case, or than CL(M) + 1, in the general case. In this weaker form, the non-degenerate Arnold conjecture has been proven for a class of symplectic manifolds known as monotone symplectic manifolds by Floer [Flo89c], then for weakly monotone symplectic manifolds by Hofer and Salamon [HS95] and Ono [Ono95] (see also [MS94] for precise definitions and a discussion on monotone and weakly monotone symplectic manifolds). So far, the conjecture for degenerate fixed points has been proven for symplectic manifold such that w and the first Chern class of TM vanish over 7r2(M) (see [Hof88, Flo89a]). In this case, the cup-length estimate implies the stronger estimate by D(M), so the Arnold conjecture holds in its original formulation (see [Rud99]). For more general symplectic manifolds, the cup-length estimate holds, provided one assumes the Hamiltonian diffeomorphism to be close to the identity in the Hofer distance (see [Sch98]).

The ideas of Kontsevich [Kon95] allowed Liu and Tian [LT98], Fukaya

and Ono [FO99] and Ruan [Rua99] to present proofs of the fact that, in the non-degenerate case, with no assumptions on the symplectic manifold, #Fix(cp) is not smaller than the sum of the rational Betti numbers of M. Clarifying certain points of these proofs and proving the same estimate in terms of the integer Betti numbers, are topics of current research.

Saddle point reductions. The reduction used in section 6.1.3 is known as saddle point reduction; it was introduced by Amann [Ama79] and further developed by Amann and Zehnder [AZ80a, AZ80b]. A simpler version of this approach can be found in Chang's book [Cha93], section IV.2.1.

Periodic points Conley conjectured that Theorem 6.3.1 should hold without any assumptions on the fixed points: any Hamiltonian diffeomorphism of (TIN,wo) should have infinitely many periodic points. The same should hold for any compact symplectic manifold (M, w) with w and the first Chern class of TM vanishing over 7r2 (M). This conjecture is still open. A positive answer to a similar problem about second order Lagrangian systems on TN has been recently given by Long [Lon00a].

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Index eigenvalues of a symplectic ma-

action functional, 75 approximation scheme, 5$ Arnold conjectures. 155. 160. 163,

trix, U eigenvalues of an infinitesimally sym-

plectic matrix, 13 equivalent inner product, 45, 48 Euler characteristic, 154 Euler functional, 84

166

asymptotically linear system, 120, 131, 1M Birkhoff-Lewis theorem, 111

Floer homology, 111 Floquet multipliers, 32, 33, 75 Fourier coefficients, S4, 66 Fourier representation, M. 72 Franks theorem, 1.29 Fredholm

Bott relations, 94 Brouwer translation theorem, 156 classical solution, 76, 84 coercive form, 51 commensurable subspaces, 44, 64 69 compact map, 57, 77 compact perturbation, 47, 52 complex projective space, 165 complex symplectic group, 11 Conley theory, 92 Conley-Zehnder index, 41 convex Hamiltonian system, 74 critical point, 78 cup-length, 93

form, 51.67 78 index, 45 index of a pair of subspaces, 49 map. 92 operator, 45

pair of subspaces, 49 62.82 function 'R, 10, 28 fundamental group of Sp(2N), 19 fundamental solution, 32

degenerate at infinity, 120,136 linear system, 38 periodic solution, 75 density of semi-simple symplectic matrices, 16 distance between subspaces, 46

generalized cohomology, 151 generalized eigenspace, 11 global twist, 133 growth condition, 75

Hamiltonian, 75 diffeomorphism, 154, 157 quadratic, 68

eigenvalue of the first kind, 19

superquadratic, 1IX1 187

Index

188

vector field, 154 Hamiltonian system convex, 74

linear and autonomous, 37 70

linear and periodic. 32, 68 nonlinear and autonomous, 75 nonlinear and periodic, Z4 on a manifold. 154 homoclinic, U8 Homology group, 91 Hopf fibration, 166 hyperbolic operator, fat index estimate, 92, 103, LIZ index pair, 93 infinitesimally symplectic matrix, 4

intersection theory of Lagrangian subspaces, 81 iteration formula, 34 35 37 iteration property of the rotation function, 21

Krein product, 52 Krein signature in Sp(2), fi in Sp(2N), 10. 12 in sp(2N), 14 Krein-negative, L 13 Krein-positive, L 13 Lagrangian, 83 quadratic. 79 Lagrangian subspace, 81 Lagrangian system linear and periodic, 78 nonlinear, 83 Landesman-Lazer condition, 127, 147

large Morse index, 53. 58 Lefschetz theory, 154

Legendre positivity condition, 80 84 Legendre transform, 79 83 Lie algebra of the symplectic group,

a linearization, 7A 84 linking, 95, 92 Marino-Prodi approximation, 144 Maslov index at infinity, 120

of a linear autonomous system, 38 of a linear system, 33 of a path, 10, 29, 31 of a periodic solution, 75 83 85 mean winding number, 35 at infinity, 120 of a linear autonomous system, 38 of a periodic solution, 75 83 minimal period, 109, 118 Morse homology, 151 Morse index, 85, ,91 Morse relations, 91 121. 127,131

negative eigenspace, 52 non-degenerate critical manifold, 94 critical point, 91 fixed point, 153 form, 51 in the autonomous sense, 131

linear system, 33 38 path, 28 periodic solution, 75, 83 normal form of a symplectic matrix, 1fi nullity at infinity, 121) of a form, 53

of a linear system, 33

Index

189

of a path, 31 of a periodic solution, 75, 83 orthogonal complement. 43 orthogonal projection. 43 Palais-Smale condition. 88 Palais-Smale-* condition, 883 12L 138

parameterization of Sp(2). 6 periodic point, 170, 172 periodic solution, Z5 Poincare polynomial. 91 Poincare's geometric theorem. 128 polar decomposition, 4. 12 polarization formula. 51 polynomial growth, 2 positive eigenspace, 52 positive form. 51 Rabinowitz condition, 141 relative dimension. 14.62 relative Morse index, Z8 of a critical point, 58 of a form. 53, 69. 81 resonant system. 33, 120. 147 rotation function

on S (2.6 on Sp(2N). 19. 21

saddle point reduction. 163 self-adjoint realization, 51 semi-simple matrix, 13 Sobolev embedding theorem. 64, 68 Sobolev spaces, 63 standard symplectic form, 2 strictly positive form, 51 strong resonance. 134. 148

subharmonic solution. 118. 130, 120 symplectic. 1

automorphism. 2

basis, 2 diffeomorphism. 1114

form. I group. 3. 4 invariant, 6. 7 21, 32 isomorphism. 2 manifold, 154 matrix. 3 subspace, 2 vector space. 1

unitary. 11 unitary group, 11

virtual critical point. 132 weakly non-degenerate, 124

MATHEMATICS

ABOUT THIS VOLUME This Research Note explores existence and multiplicity questions for periodic solutions of first order, non-convex Hamiltonian systems. It introduces a new Morse index theory that is easier to use, less technical. and more flexible than existing theories and features techniques and results that, until now, have appeared only in scattered journals. In six succinct chapters. the author provides a self-contained treatment with full proofs. The purely abstract functional aspects have been clearly separated from the applications to Hamiltonian systems, so many of the results can be applied in other areas of current research, such as wave equations. Chern-Simon functionals, and Lorentzian geometry. Morse theory for Hamiltonian systems not only offers clear, well-written prose and a unified account of result.-, and techniques. but it also stimulates curiosity by leading readers into the fascinating world of symplectic topology.

Readership: This volume will prove particularly interesting to mathematicians working in nonlinear analysis. calculus of variations. Hamiltonian systems. and symplectic geometry.

CHAPMAN & HAWCRC RESEARCH NOTES IN MATHEMATICS SERIES The aim of this series is to disseminate important new material of a specialist nature in economic form. It ranges over the whole spectrum of mathematics and also reflects the changing momentum of dialogue between hitherto distinct areas of pure and applied parts of the discipline. The editorial board has been chosen accordingly and will, from time to time, be recomposed to represent the full diversity of mathematics covered by Mathematical Reviews. This is a rapid means of publication for current material whose style of exposition is that of a developing subject. Work that is in most respects final and definitive, but not yet refined into a formal monograph. will also be considered for a place in the series. Normally, homogeneous

material is required. even if written by more than one author; thus multi-author works will be included provided there is a strong linking theme or editorial pattern.

Proposals and manuscripts: See inside book

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Morse Theory for Hamiltonian Systems

Morse theory for Hamiltonian systems

Morse theory of completely integrable Hamiltonian systems

Morse Theory

Critical Point Theory and Hamiltonian Systems

Circle-Valued Morse Theory

Multi-Hamiltonian theory of dynamical systems

Critical Point Theory and Hamiltonian Systems

Four Hamiltonian Systems

Circle-valued Morse theory

Invitation to Morse Theory

Instability in Hamiltonian systems

Integrable Hamiltonian systems

Symplectic Geometric Algorithms for Hamiltonian Systems

Morse Theory. (Am-51)

Stratified Morse theory

A geometric setting for Hamiltonian perturbation theory

An introduction to Morse theory

Hamiltonian Dynamical Systems and Applications

Variational and Hamiltonian Control Systems

An Invitation to Morse Theory

Soliton equations and Hamiltonian systems

Hamiltonian dynamical systems and applications

Hamiltonian Dynamics Theory And Applications

An invitation to Morse theory

Hamiltonian Dynamics Theory and Applications

Introduction to the Perturbation Theory of Hamiltonian Systems

Introduction to the perturbation theory of Hamiltonian systems

Differential Galois Theory And Nonintegrability Of Hamiltonian Systems

Critical point theory and Hamiltonian systems (Applied Mathematical Sciences)

Hamiltonian dynamics: theory and applications

Morse Theory for Hamiltonian Systems