Qualitative theory of foliations on closed surfaces

Jmtrnal c)f Mathenuttical Sciemes. V.I. 90. No. 3, 1998

QUALITATIVE THEORY

OF FOLIATIONS ON CLOSED SURFACES

S. Kh. A r a n s o n , I. U. B r o n s t e i n , E. V. Z h u z h o m a , and I . V . N i k o l a e v

UDC 517.987.5; 517.933

Preface Foliations on surfaces are natural generalizations of flows (continuous-time dynamical systems) on surfaces. According to the classical theory, in the neighborhood of a nonsingular point (i.e., a point different from the equilibrium state), the trajectories of a flow are a family of parallel straight lines. This is what served as a starting point for defining a foliation. Let M be a closed surface (a closed two-dimensional manifold). The foliation F on M with a set of singularities Sing (F) is the partitioning M - Sing (F) into nonintersecting curves (called fibers) which are locally homeomorphic to the family of straight lines. As to the set of singularities Sing (F), this set must be described separately for every class of fibers. In the sequel, unless otherwise specified, we shall consider the set Sing (F) to be finite. Foliations occupy an intermediate place between flows and arbitrary families of curves on surfaces. Foliations which can be embedded into a flow are said to be orientable; otherwise they are nonorientable. The origination of a qualitative theory of foliations goes back to the works by H. Kneser, G. Reeb, A. Haefliger, and S. P. Novikov. The theory of foliations attracted special attention early in the sixties in connection with the study of Y-flows and Y-diffeomorphisms introduced by D. V. Anosov. The technique of foliations allowed G. Franks to classify Y-diffeomorphisms of codimension one the nonwandering set of whose points coincides with the whole manifold. The application of a "surgical operation" to Y-diffeomorphisms of codimension one leads to nontrivial base sets of codimension one (attractors and repellers) the profound results in the study of whose geometry and topology belong to P. V. Plykin and V. Z. Grines. A more general approach was suggested by Ya. G. Sinai, who introduced a class of dynamical systems with two invariant transversal fibers. When we use this approach, we neglect such properties of the systems as the estimates of the contraction and extension of the fibers of invariant foliations, the everywhere density of the periodic trajectories in a nonwandering set, and others, and leave only those properties which are closely connected with the topology' of the manifolds. A new impetus to the study of foliations and homeomorphisms with invariant foliations was given by the works of W. Thurston, in which he complemented the homotopic classification of homeomorphisms of surfaces obtained by d. Nielson in 1920-30 and gave a new interpretation to it. The introduction by Thurston of the concept of pseudo-Anosov homeomorphism, which generalized the concept of the Anosov diffeomorphism, stimulated further investigations in this direction based on the study of the action of homeomorphisms in a fundamental group. Beginning in the 1980s, the geometry and topology of foliations with saddle-point singularities were studied by H. Rosenberg (the construction of labyrinths), G. Levitt (equivalence in the sense of Whitehead), G. Papandopoulos, K. Dantoni, and others. At the same time, many questions concerning the qualitative theory of foliations (the topological and smooth classifications, the structural stability, typicalness and so on) remained open. This review is devoted to the contemporary state of the qualitative theory of foliations with singularities on closed surfaces, tile main emphasis being placed on the development of the theory of Poincard-Bendixon Translated from ltogi Nauki i Tekhniki, Seriya Sovreme,l,mya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 32, Dynanlical Systems-5, 1996. 1072-3374/98/9003-2111520.00 9

Plenum Publishing Corporation

2111

for foliations (Chap. 1), the local theory (Chap. 2), the theory of stability (Chap. 3), and the topological classification (Chap. 4). For flows and cascades on surfaces, similar problems were considered in reviews [7, 8] and in the monographs [1, 2] (see also the references in these works).

2112

Chapter 1 C E R T A I N A S P E C T S OF T H E Q U A L I T A T I V E T H E O R Y O F FOLIATIONS ON SURFACES

Introduction The qualitative theory of nonorientable foliations (foliations which cannot be immersed into a flow) followed a different scenario (opposite, in a sense) than the development of the qualitative theory of flows. The local theory, the bifurcation theory, and the theory of stability were always dominating in the theory of flows. This was due to the fact that, in accordance with the Poincard-Bendixon theory, there are no nonclosed Poisson stable semitrajectories in the plane or a plane domain, and the list of limit sets for flows with a finite number of equilibrium states is exhausted by equilibrium states, closed trajectories, and separatrix contours. In addition, smooth flows are defined by vector fields which are studied with the use of a powerful apparatus of integral and differential calculi. In 1937, A. A. Andronov and L. S. Pontryagin obtained a list of structurally stable singular trajectories of a flow within a cycle without a contact in a plane domain and established the necessary and sufficient conditions for roughness (structural stability) of flows of this kind. In the 1950s, E. A. Leontovich and A. G. Mayer found a complete invariant, which they called a scheme, defining the topological structure of a flow on a plane domain with a normal boundary and having a finite number of singular trajectories (equilibrium states, limit cycles, and separatrices). This (locally, in essense) theory culminates in the 1960s in M. Peixoto's classification of structurally stable flows (they proved to be Morse-Smale flows on an orientable surface) on closed surfaces. In the late 1930s and early 1940s, T. M. Cherry and A. G. Mayer described the types of trajectories of a quasiminimal set, i.e., a set which is the closure of a nonclosed Poisson stable semitrajectory, and of a nontrivial minimal set, i.e., a set which is the closure of a nonclosed recurrent semitrajectory. They proved a number of remarkable theorems on the structure of these sets. The study of nontrivial minimal sets and quasiminimal sets of flows on a two-dimensional torus was carried out in the works of Poincar4, Denjoy, Cherry, and others. A sufficiently complete investigation of these sets was reduced to a one-dimensional dynamics, i.e., the study of the transformations of a circle. Only recently, due to the efforts of D. V. Anosov, S. Kh. Aranson, V. Z. Grines, W. Gardiner, G. Levitt and some other mathematicians, was some progress attained in the study of the geometry and topology of flows with complicated invariant sets on closed surfaces of any kind. Aranson and Grines classified (in the topological sense) transitive flows and nontrivial minimal sets of flows on orientable closed surfaces of a negative Euler characteristic (hyperbolic surfaces). Gardiner and Levitt proved that a hyperbolic surface, on which a flow is defined, could be cut by simple closed curves, nonhomotopic to zero, so that on every resulting compact two-dimensional manifold the flow either does not have any nonclosed Poisson stable semitrajectories or has exactly one quasiminimal set. In contrast to the orientable case, nonorientable foliations with nonclosed Poisson stable semifibers can exist in a disk and, consequently, on any plane domain. In addition, only recently was a sufficiently acceptable apparatus worked out for studying nonorientable singularities of foliations. Aranson and Zhuzhoma generalized the classical theorems of Cherry and Mayer for nonorientable foliations with a finite number of singularities. For foliations with saddle singularities with a negative index on 2113

Fig. 1.1 orientable hyperbolic surfaces, Levitt proved the possibility of cutting the surfaces by closed transversals on domains without nonclosed Poisson stable semifibers and on domains with one quasiminimal set. Aranson has obtained a topological classification of transitive foliations and nontrivial minimal sets of foliations on closed hyperbolic surfaces. Bronstein and Nikolaev constructed a local theory of the structural stability of nonorientable singularities of foliations, and this allowed them to isolate a class of structurally stable foliations (similar to the class of Morse-Smale flows) and to classify them.

1.1.

Main Definitions and Examples

Suppose that M is a closed surface and F is a foliation on M. A fiber of the foliation F is closed if it is homeomorphic to a circle. Let l be a nonclosed fiber. Any point m E l cuts l into components 11,12, each of which, with an added point rn, is called a semifiber with initial point ra. On any semifiber l() with initial point z we can introduce the parametrization t: l() ---+ [0, +oo), t(m) = 0, such that t is a homeomorphism in the internal topology of the semifiber l ( ) . The limit set lim(l()) = fl(l()) of the semifiber l() is a set of points in any neighborhood of which there exist points of l() with infinitely large values of the parameter. Obviously, the set lira(/(l) on a closed surface is nonempty, closed, and invariant. (i.e., consists of whole fibers and singularities). The semifiber l() of a nonclosed fiber is said to be nonclosed, Poisson stable if l() C lira(l()). This is equivalent to the equality clos (l()) = lim(l()), where clos (.) means the operation of closure in the set-theoretic sense. A nonclosed fiber is nonclosed, Poisson stable if its two semifibers are Poisson stable. The closure of a nonclosed Poisson stable semifiber is known as a quasiminimal set of a foliation. Here are some examples of foliations and singularities of foliations. 1. Let f t be a flow with isolated equilibrium states fix ( f ' ) on the surface M. T h e integral curves of the flow f t form a foliation F with Sing (F) = fix ( f ' ) on M. The foliation F is orientable. 2. The simplest nonorientable foliation on a closed orientable surface is a foliation without singularities on a two-dimensional torus containing the so-called Reeb component (see Fig. 1.1). 3. Stable and nonstable manifolds of pseudo-Anosov and generalized pseudo-Anosov homeomorphisms of surfaces form transitive foliations with singularities which are topological saddles. Recall that a topological saddle with u separatrices is a singularity of a foliation in whose neighborhood the foliation is defined by the equation (x + iy)" = (t + ic) 2 if u is odd or by the equation (x + iy) ~ = t + ic if u is even, where x , y are local coordinates in the neighborhood of the singularity (z = 0, y = 0), t is a parameter, c = const, i 2114

Fig. 1.2

'

--5

Fig. 1.3 is an imaginary unity (Fig. 1.2). Let Sep (O) be a family of curves passing through the point (0,0). The components of the set Sep (O) - (0, 0) are called separatrices. A saddle with one separatrix is called a thorn, a saddle with two separatrices is known as a fake saddle, that with three separatrices is called a tripod and that with four separatrices an ordinary saddle. Note that for the singularities of the invariant foliations of pseudo-Anosov and generalized pseudo-Anosov homeomorphisms, v -~ 2. A significant concept of the theory of foliations is the concept of an index of an isolated singularity. Suppose that F is a foliation on M and C is a simple closed curve, bounding the domain D, which is homeomorphic to a disk. We assume that C is transversal (in the topological sense) to F , except for a finite number of points. We denote by I (respectively J) the number of points of external (internal) contact of the fibers of the foliation F with C with respect to the domain D. The indez of the closed curve C is the number

I-J

ind ( C ) = 1 - - - ~

Note that by a small perturbation we can reduce any closed curve on a surface to a general position with respect to the foliation (i.e., to a curve with a finite number of contacts with F). It immediately follows from the definition of an index that it is either integral or semi-integral. If the disk D contains exactly one singularity So, then we can show that the q u a n t i t y ind (C) does not depend on the choice of C. The number ind (C) is known as the indez of the singularity So and is denoted by ind (So). For instance, the indices of a thorn, a fake saddle, a tripod, and an ordinary saddle are equal, respectively, to 89 0, - ~ , - i , and the index of any saddle with u separatrices is equal to 1 - ~. It is known that for foliations with a finite number of singularities the sum of indices of all singularities is equal to the Euler characteristic of the surface. Two foliations FI, F2 are topologically equivalent if there exists a homeomorphism ~5: M --* M which maps the fibers of the foliation /71 into those of the foliation F2, with r (/71)) = Sing (F2). Two foliations F1, F2 on M are Whitehead equivalent if one of them carl be obtained from the other by means of the operations shown in Fig. 1.3. 2115

Fig. 1.4 1.2.

T h e o r e m s of t h e C h e r r y a n d M a y e r T y p e for F o l i a t i o n s

At the end of the 1930s and the beginning of the 1940s, Cherry and Mayer obtained classical results concerning the structure of a quasiminimal set of orientable foliations with an arbitrary collection of singularities on surfaces [33, 21]. Cherry proved that in any quasiminimal set there existed a continuum of everywhere dense in it nonclosed Poisson stable fibers and described possible types of fibers (note that Cherry's theorem is valid for flows on n-dimensional manifolds satisfying the second axiom of countability). Mayer established, for compact orientable surfaces, that in a quasiminimal set of an orientable foliation, any nonclosed Poisson stable semifiber contained in it was everywhere dense. He also obtained the necessary and sufficient conditions for a nonclosed fiber to be a nonclosed Poisson stable semifiber. Aranson and Zhuzhoma generalized these theorems by Cherry and Mayer to nonorientable foliations (an analog of Cherry's theorem is valid for foliations with any, possibly infinite, set of singularities) [12] and showed that the analog of Mayer's theorem for nonorientable foliations with an infinite number of singularities was, in general, not satisfied. The following theorem describes possible limit sets of a separate semifiber of any foliation with a finite number of singularities on a closed surface and is a generalization of the Poincar~--Bendixon theory for flows (in fact, orientable foliations) on the plane. T h e o r e m 1.1. Suppose that a foliation F with a -finite number of singularities is defined on the closed surface M and that l is a nonclosed semifiber of the foliation F. Then the limit set lira(l) of the semifiber l can only be of one of the following types: (1) lim(l) is ezactly one singularity; (2) lira(t) is ezactly one closed fiber; (3) lira(l) is ezaclly one closed contour consistin 9 of singularities and separatriz connections, and in the case of a -finite number of separatrix connections every separatriz connection is a Bendizon continuation of the preceding separatriz connection (for a definite transition of the closed contour, Fig. 1.4); (4) lira(l) is a quasiminimal set. R e m a r k 1.1. The concepts of a separatrix of an arbitrary isolated singularity of a foliation and of its Bendixon continuation are completely similar to the corresponding concepts for flows [1]. Here is an analog of Cherry's theorem. T h e o r e m 1.2 ([12]). Let Q be a quasiminimal set of the foliation F on the closed surface M. contains a continuum of nonclosed Poisson stable fibers, each of which is everywhere dense in Q.

Then Q

R e m a r k 1.2. Levitt [44] proved this theorem for foliations with saddle singularities with a nonzero index defined on orientable surfaccz: with the additional assumption that each fiber from Q, which is not a separatrix connection, is everywhere dense in Q. K. Yano proved Theorem 1.2 for foliations of any dimension (a quasiminimal set is defined as the closure of an improper fiber) on any manifold satisfying the second axiom of countability (private communication). 2116

The following two theorems are analogs of Mayer's theorem. The first of them gives the Poisson stability criterion of a nonclosed semifiber. T h e o r e m 1.3 ([12]). Suppose that a foliation closed surface M. We assume that the limit set fiber which is different from the singularity and l, foliation F. Then l is a nonclosed Poisson stable C o r o l l a r y 1.1. Suppose that a foliation F surface M and that Q is a quasiminimal set limit set different from one of the singularities nonclosed Poisson stable semifiber everywhere

F with a -finite number of singularities is defined on the of the nonclosed semi-fiber l of the foliation F contains a in turn, belongs to the limit set of a certain semifiber of the semifiber.

with a finite number of singularities is defined on the closed of the foliation F. Assume that the semifiber l C Q has a (i.e., l is not a semiseparatriz of the singularity). Then l is a dense in Q.

T h e o r e m 1.4 ([12]). Suppose that a foliation F with a finite number of singularities is defined on the closed surface M, and assume that ll,l~ are nonclosed Poisson stable semifibers. Then, if ll C lim(12), then 12 C lim(lx). Foliations with a continual set of singularities for which Theorems 1.3 and 1.4 are not valid were constructed in [12]. In particular, for the foliation F with an infnite number of singularities, a situation can occur where one quasiminimal set is a proper part of the other. In this connection we give the following definition. The sequence of proper inclusions

Q~ c Q~ c . . . c Qk, where all Q; are quasiminimal sets of the foliation F, is called a nest of length k. The maximal length of various nests is known as the depth of the foliation F. By virtue of Theorem 1.4, any foliation with a finite number of singularities on a closed surface has a depth not exceeding 1. For a foliation with an arbitrary set of singularities, the following conjecture seems to be valid. C o n j e c t u r e . On a closed surface any foliation has a finite depth.

1.3.

T h e S t r u c t u r e of a Q u a s i m i n i m a l S e t

We denote by C H ( F ) the union of all nonclosed Poisson stable fibers and semifibers of the foliation F. This set contains fibers with a very "chaotic" behavior. For an orientable foliation F on a closed surface, the set-theoretic closure clos (CH(F)) of this set is a finite union of quasiminimal sets which can be pairwise intersecting only along separatrix contours and singularities [21, 3]. For nonorientable foliations this result is valid under the conditions of a finite number of singularities. For foliations with an arbitrary set of singularities this result is, in general, not valid [12]. T h e o r e m 1.5. Suppose a foliation F with a finite number of singularities is defined on the closed surface M and let C H ( F ) r 9. Then clos (CH(F)) is a finite union of quasiminimal sets. Two quasiminimal sets can intersect only along separatriz contours and singularities. For foliations which have only saddle singularities, Theorem 1.5 was proved on an orientable surface by Levitt [44]. In the general form it was proved by Aranson and Zhuzhoma [12]. For orientable foliations with an arbitrary set of singularities, the maximal number of quasiminimal sets depends only on the genus p of the surface; it does not exceed p if the closed surface is orientable [21] and does not exceed the integral part of 0.5(p - 1) if the closed surface is nonorientable [3, 48]. In contrast to orientable foliations, nonorientable foliations can have any preassigned number of quasiminimal sets on any surface (including a sphere and a projective plane). Therefore, the upper estimate of the number of quasirninimat sets of nonorientable foliations depends not only on the topological structure of the 2117

surface but also on the embedding of the quasiminimal sets in the surface. The e x p o n e n t characterizing this embedding is the thorn index of the foliation. Let us move onto exact definitions. Suppose that F is a foliation on the surface M and E is a line segment w i t h o u t contact or a closed transversal (in the topological sense) for the foliation F. We assume that the fiber 1 of the foliation F cuts E at the points a,b. The arc abof the fiber 1 between the points a and b is known as a E - a r c if~-gfqE = {a}U{b}. Let Q be a quasiminimal set of the foliation F with a finite number of singularities Sing (F). We assume that Q is cut by at least one closed curve of positive semi-integer index which b o u n d s a simply connected domain from Q on the surface. Then there exists a subset of singularities or(Q) C_ Sing (Q) which satisfies the following properties. 1. If a contactless segment E and a E-arc of the fiber l C Q bound a disk on M, then this disk contains at least one singularity from (r(Q). 2. For every singularity So C or(Q) there exist a contactless segment A and a A-arc of a certain fiber l' C Q which bound the disk, and this disk contains exactly one singularity So from t h e set ~r(Q). The actual composition of the set or(Q) is, in general, defined not uniquely but its power t(Q) depends only on the immersion of the quasiminimal set Q in M. The number t(Q) is known.as the thorn index of the

quasiminimal set Q,. If all the singularities in Q have an integral index, then we consider the thorn index to be equal to zero. The sum of thorn indices of all quasiminimal sets of the foliation F is called the thorn index of the

foliation F. 1.6 ([12]). Suppose that a foliation F with a finite number of singularities and thorn index t(F) is defined on the closed surface M of genus p and let N ( F ) be the number of quasiminimal sets of the foliation F. Then (1) if the surface M is orientable, then

Theorem

N ( F ) < p + max'r~ [P+ -

t(F)

1];0} def = or (p, t),

2

where [-] is the integral part; (2) if the surface M is nonorientable, then N(F)

< max

{[ap+ 2t(s)5] ;0 } .o,: n o r ( p , t ) ; 2

(3) for any integer t > 0 on the surface M there exists a foliation F with a finite number of singularities

and with thorn index t = t( F) which has exactly N(F)

f or (p, t) nor (p, t)

if the surface M is orientable, if the surface M is nonorientable

quasiminimal sets. In order to prove Theorem 1.6, we use the theorem on the existence of a closed simple transversal (in the topological sense), which is of interest in itself. 1.7 ([12]). Suppose that a foliation F with a finite number of singularities is defined on the closed surface M and let Q be a quasiminimal set of the foliation F. Then there exists a simple closed transversal of the foliation F which cuts Q and does not cut any other quasiminimal sets of the foliation F.

Theorem

Note that the estimate of the number of quasiminimal sets for foliations with only saddle singularities on an orientable surface was obtained by Levitt [44]. If the thorn index of a quasiminirnal set is zero, then this quasiminimal set is said to be exteriorly situated. 2118

Note that a foliation with exteriorly situated quasiminimal sets can be nonorientable. Moreover, on a closed orientable surface of genus 3 we can construct a nonorientable foliation w i t h one quasiminimal set which is exteriorly situated and is such that all singularities of the foliation are o r d i n a r y saddles. The concept of exterior situation was introduced by Ptykin [23] for one-dimensional base sets of Adiffeomorphisms of surfaces. We have used this term since a one-dimensional base set can be immersed in a foliation for which this base set is a quasiminimal set and, in addition, the exteriorly situated one-dimensional base set corresponds to the exteriorly situated quasiminimal set. Theorem 1.7 yields estimates of the number of exteriorly situated quasiminimal sets (for t(F) = 0) of foliations on orientable and nonorientable closed surfaces. In particular, we find t h a t on a sphere, a projective plane, and the Klein bottle, no foliation with a finite number of singularities can have exteriorly situated quasiminimal sets.

1.4.

D e c o m p o s i t i o n of F o l i a t i o n s i n t o R e c u r r e n t D o m a i n s a n d P a s s i n g D o m a i n s

Everywhere in this section M is a closed orientable surface and F is a foliation on M which has singularities only of saddle type. A foliation is said to be arationaI if it does not have closed fibers, separatrix cycles, or separatrix connections of the type thorn-thorn, thorn-saddle. T h e o r e m 1.8 ([44, 45]). Let F be an arational foliation on M. Then there exists a foliation F', equivalent to the foliation F in the Whitehead sense, such that the following conditions are satisfied. 1. The foliation F' has a family E of pairwise nonintersecting closed transversals which do not intersect with the quasiminimal sets of the foliation F'. 2. Every component of the set M - E contains not more than one quasiminimal set of the foliation F'. The closure of the component of the set M - E which contains (does not conta.in, resp.) a quasiminimal set, is called a recurrent domain (passing domain, resp.). The pair ( F ~, E) satisfying Theorem 1.8 is known as the Levitt decomposition of the foliation F if every transversal from E, within the boundary of the domain of permeability, also enters into the boundary of the recurrent domain. The Levitt decomposition is not uniquely defined by the initial foliation. However, from all decompositions we can isolate the simplest one in a certain sense. The multiplicity of the saddle with u separatrices is the number u - 2 (i.e., the multiplicity is the number of tripods into which the saddle can be decomposed by means of Whitehead's operation). Let us assign to the Levitt decomposition ( F ' , E ) the pair of numbers (s,g), where s is the sum of multiplicities of all saddles lying in the domain of recursiveness and g is the sum of genuses of all domains of recursiveness. Then we introduce a lexicographical order on the pairs (s,g). The Levitt decomposition with minimal pair (s,g) is minimal. T h e o r e m 1.9 ([44, 45]). Let (F1, El), (1:2, E2) be two minimal Levitt decompositions of the arational foliation F. If any separatrix connection of the foliation Fi cuts a transversal from Ei (i = 1,2), then there exists a homeomorphism of the surface, isotopic to the identical one, which maps E1 into El and carries out a topological equivalence of the foliations 1:1, Fl. Thus the minimal decomposition in the sense of Levitt is defined uniquely, in some sense. Anton Zorich [58] proved a theorem on the decomposition of an orientable foliation defined by a closed 1-form of maximal rank. We shall present the result in greater detail. Suppose that a closed 1-form w which has a finite number of nondegenerate critical points is defined on the closed orientable surface .~,tp of genus p > 2. We also asume that different critical points lie at different critical levels and the form ca is of the maximal possible rank 2p, i.e., all periods of the form w are rationally 2119

independent. By analogy with the terminology used in [58], the foliation F ~ form to, is said to be Hamiltonian.

F(to) on alp, defined by the

T h e o r e m 1.10 ([58]). Suppose that the closed form to defines the Hamiltonian foliation F(co) on the closed orienlable surface Mp of genus p > 2. Then there ezists a decomposition Alp into a finite number of compact submanifolds, filled up by closed fibers of the foliation F(to), and a finite number of compact submanifolds, M p ~ , . . . , Mp~, that satisfy the following properties. 1. The surface Mpi has genus pi (i = 1 , . . . , k ) , and Pl + "'" + Pk = p. 2. Any (one-dimensional) fiber of the foliation F(to) lying inside M m is everywhere dense in M m (i =

1,... ,k). 3. The bounding of the foliation F(to) on every M m is an orientable foliation immersed in the .flow f~, which has a closed global secant C; (i = 1 , . . . , k ) . In this case f~ is a special flow over the ezchange transformation of the intervals 4pi - 4 lying in Ci.

It is also shown in [58] that for the typical form to the above-indicated interval exchange transformations are strictly ergodic, the ergodic measure being defined by the bounding of the form w on Ci (for definitions of the corresponding notions of the ergodic theory, see [19]).

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Chapter 2 LOCAL THEORY

Introduction A smooth vector field on a manifold defines a phase pattern (i.e., a one-dimensional foliation, possibly with singularities). It was noticed long ago that there existed foliations with singularities in the plane which cannot be represented as phase patterns of a continuous vector field since they are nonorientable in the neighborhood of a singularity (see, for instance, [22, p. 45], where the so-called tripod is described). For the investigation of one-dimensional foliations with nonorientable singularities Guti6rrez [39], Kadyrov [18], Bronstein and Nikolaev [16, 17, 31, 32, 50, 51] suggested the use of the concept of the field of line elements. In these works, they understood line elements as the pair { v , - v } of mutually opposite tangent vectors. In the theory of foliations with nonorientable singularities the fields of line elements play the same part that vector fields play in the theory of locally orientable foliations. In a short paper [39], Guti6rrez made a not quite correct attempt to reduce the field of line elements to a pair of mutually opposite vector fields defined in the same plane. We could not find a detailed exposition of these results. For studying the fields of line elements {(vl, v2),-(vl, v2)} in the plane, Kadyrov [18] suggests the consideration of the field (v2a - v~, 2v~v2), which, however, is not a vector field. In their works, Bronstein and Nikolaev develop an approach to the investigation of the fields of line elements based on the Riemann classical construction. They show that such a field, defined on the plane P, can be doubly covered by a vector field on the auxiliary plane Q with a ramification at a singular point. This natural approach makes it possible to describe the typical singularities of the fields of line elements on the plane and establish their structural stability. It is also shown that the field of line elements on a compact two-dimensional manifold can be reduced to a vector field on a doubly (ramified) covering manifold which satisfies the condition of antisymmetry relative some smooth involution. In this way, they obtained a generalization of the well-known Andronov-Pontryagin theorem to the case of fields of line elements. - Some problems, directly or indirectly connected with the fields of line elements, considered in the works of Guinez [37, 38], Guti6rrez and Sotomayor [40, 41], and Michel [49], are devoted to the geometry of positive definite differential 2-forms and curves of smooth curvature of the Riemannian surfa.ce. The geometrical properties of foliations and laminations with singularities on a surfaces were investigated by Thurston [57], Levitt and 1-(osenberg [55, 44-47], Aranson and Zhuzhoma [12-14, 4, 5, 30]. Note that foliations with singularities are encountered in physical studies (for instance, in the theory of liquid crystals as the disclination of the crystal structure of nematics, see [20, 27]). Kadyrov discusses in [18] the connection between the fields of line elements and plane deformations of elastic plates.

2.1.

Preliminaries

Suppose that 31 is a smooth n-dimensional manifold without boundary and ( T M , rM, M) is its tangent foliation. Let us determine in 7'5'1 the equivalence relation E: El (2 if r^t(~l) = rgt(~2) and (, = -l-G2. We denote by L M the quotient space T M / E and by A the natural projection LM ~ M. The fiber L~M "~ T~M ""

2121

of the foliation ( L M , A, M) over x E M is a cone o v e r / { p ~ - l . In the two-dimensional case, the fiber L,:M is homeomorphic to t{2. The continuous mapping a: M ~ L M , which satisfies the condition A o a = idM, is known as a field of line elements. It should be empl',asized that even in the case of dim M = 2, there is no natural differentiable structure on L:~M and, consequently, on L M . Thus the concept of the smoothness of a field of line elements cannot be defined in this way. In particular, it is meaningless to speak of the structural stability of the fields of line elements or: M -o L M as such. The main point of view accepted in [31] consists in comparing the field o~: M ---+ L M and the vector field v: M ---+T M defined on a doubly covering manifold M- and in investigating problems concerning fields of line elements in terms of the corresponding vector fields. We realize t h a t this approach m a y seem to be not very satisfactory and even somewhat artificial, but we do not see any reasonable alternative. The advantages of the suggested approach consist in the possibility of widely using analytical methods. For other definitions of the differentiability of foliations (and the corresponding field of line elements) in the neighborhood of a singular point, see [44, 47]. We think that the definitions introduced there are artificial since they are not suitable for the investigation of foliations and are based on the "transversality" of smooth fibers. If the manifold M is simply connected and a ( M ) does not intersect the zero section of the foliation L M , then the field of line elements is equivalent to the pair of mutually opposite vector fields vx, v2: M --+ T M . The singularity of the field of line elements c, is trivial if the field a in the deleted neighborhood of the singular point reduces to a pair of mutually opposite vector fields. Let us consider the field of line elements a:/{2 __0 TR2 such that a is zero at a single point (say, at the origin), i.e., a(O) = 0 = (0,0), ~(z) • 0 (z E /{2\{0}). Let us recall the classical construction dating back to Riemann. Let 1 be a ray e m a n a t i n g from the origin. Since R=\I is an open simply connected domain and a(x) -~ 0 for x E/{2\1, the field of line elements a defines the pair of continuous vector fields Vl and v= which are defined in R2\I and are such t h a t vx + v= = 0. Let us map diffeomorphically the domain R \ I onto the open half-plane /{~. of a certain auxiliary plane. This

diffeomorphism maps the vector field vl into the plane R~.. Taking the composition of this diffeomorphism with rotation through the angle 7r, we project the vector field vl into the lower half-plane R 2- and extend it by continuity to the closure ~2_. Gluing the half-planes R~. and ~2_ together, we arrive at a continuous vector field ~[ on R ~ satisfying the following condition of antisymmetry: {(x) + sc(-z) = 0, z E R 2, {(0) = 0. Conversely, let {: i{2 __+ T/{~ be a continuous vector field with the a n t i s y m m e t r y indicated above. We denote by p:/{2 __+ R2 the mapping which glues the points (z, y) and ( - x , - y ) into one point and which is a ramified double covering. Clearly, the vector field { defines (by means of the mapping p) the field of line elements a = a(x) on p ( R 2) = /{2. If we write the vector field {: R 2 ---, T/{ 2 as the system of differential equations

dz

7/=

P(z,y),

dy

d-/=

P(0,0) = Q(0,0) = 0,

then the above condition of antisymmetry means that P ( z , y) = P ( - x , - y ) ,

Q(z, y) = Q ( - z , - y ) .

The a n t i s y m m e t r y condition {(x) + ~ ( - x ) = 0 is not invariant with respect to the change of coordinates. However, it can be reformulated in an invariant form, namely, the vector field ~: R 2 ---+ T R 2 must satisfy the condition of a n t i s y m m e t r y with respect to the C ~ - s m o o t h (preserving the orientation) local diffeomorphism j:/{2 __} R 2 with j2 = id, J(O) = O, DJ(O) = - i d , D J ( { o J) = -,{. (2) Lemma 2122

2.1.

The involution J is a formal conjugate of the rotation through the angle 7r.

Proof. Let us consider the germ of the diffeomorphism J: R`2 ~ R 2 at the point 0 and assume that the jet of order N of the map J + id is zero. We shall prove by induction that the (N + 1)th jet of this sum is also zero. Two cases are possible here. (1) N + 1 is an even number, N + 1 = 2n. The diffeomorphism J : ( z , y ) --+ ( u , v ) has the form k

v~ (k) k-t t L at z y ,

u = -z + k:2n r

/=0 k V"

v = -y +

(3) l

k=2n /=0

It is easy to verify that the polynomial change of coordinates 2n

x

(2,~)

xl +

=

a

2n-I I x 1 Yl,

l=0 2n

(4) l(2n)

Y = Yl + 2.., o

2n-I

I

z1

Yl

1=.0

normalizes the 2nth jet, i.e., after transformation (4) we obtain al 2n) = bl2") = O, (l = O, 1 , . . . , 2n). (2) N + 1 is an odd number, N = 2n. In this case, the condition j2 = id imposed on the germ (3) implies that al 2n+') = bl2'~+') = 0, (l = O, 1 , . . . , 2 n + 1). It follows from this lemma that the involution J is C a - s m o o t h l y equivalent to a diffeomorphism of the form - i d + r where r is an additional component which is plane at the origin. We do not know, however, whether it is possible to linearize J in the class of C~-diffeomorphisms. Thus, in the typical case, the study of fields of line elements in the neighborhood of an isolated nontrivial singular point reduces to the study of the vector field ( 1 ) a n d the involution J with properties (2). The field of line elements a: R 2 ~ T R 2 is Ck-smooth if the corresponding vector field ~: R 2 ---* T R 2 belongs to the class C k. Moreover, different topologies in the space of antisymmetric vector fields generate corresponding topologies in the space of fields of line elements.

2.2.

I n v e s t i g a t i o n of F i e l d s o f L i n e E l e m e n t s in t h e N e i g h b o r h o o d o f a N o n t r i v i a l S i n g u l a r i t y

We denote by F the set of vector fields (1) which satisfy conditions (2) and which are C2-smooth in the neighborhood of the origin. Every vector field ~ E F can be written as dx

d-7 =

P 2 ( x , y) + r

y),

dy

=

y) +

where P2 and Q`2 are homogeneous polynomials of degree 2 and r z2 + y2 ___,0. The system of the first approximation dx 1 `2 2a~`2xy 1 `2 d-[ : a l l x + + a22Y '

dy

d--/

y)'

(5)

y) = o(x 2 + y~), r

y) = o ( x 2 + y2) as

2 x`2 2a~2xy `2 2 all + + a22Y

:

(6)

has been thoroughly investigated (see, for instance, [28]). We introduce the notation r

1

2

2

. -27(a`22). (a,,)

2

:2

1 1 .4a~2(a, :2

2

3

l

2

l

2

.2a1`2) + 18a`2`2all (a,l

1

3

1

2

2

:2

2

2a12)(a2~ ,'~

l 2a12)

1 ',`2

--4a,,(a2`2 _ 2a,`2) + (a,, __ 2a12 ) (a22 -- z.al2 ) , #

=

1 "2 (a~a22

-

1 2 ",:2 a2..,al2 )

-

-

1 2 -l(allal2

-

l 2 al2all

I

2

)(a,2a22

l 2 -- a2:2a12 ) .

2123

We denote by 1'o a subspace of I" consisting of vector fields that satisfy the condition r] r 0, # # 0. It is clear that ['0 is open and everywhere dense in the space [" provided with the C2-topology. For all vector fields E F0 the origin is an isolated singular point. As follows from the results of [28], system (5) with r/ r 0, # :~ 0 can be reduced (by means of a linear nondegenerate transformation) to one of the following normal forms:

dx

- - = a 2 + I ~ x y - Y2 + C(z,Y), dt

dy

- - : (a + I)zy + I3Y~ + r 2; Y), dt '

= a[Z ~ + (a + 1)~1 r 0,

(7) (8)

or

dx

d--7 = ~2;= + (fl - 1)xy + r

dy

y),

d~ =

(a - 1)xy + fly2 + ~b(x, y),

~, = ~ # ( ~ + ~ - 1) r 0.

(9)

(10)

L e m m a 2.2. The phase patterns of vector fields ~ E F0 in the neighborhood of the origin are exhausted by the phase patterns with the numbers 1, 2, 3, and 5 shown in Fig. 2.1 (with an accuracy to within the topological equivalence). The vector fields ~ E Fo in the neighborhood of the origin are C2-structurally stable relative to the perturbations from F. Proof. We begin by studying system (7) with condition (8). We shall use the blow-up (solution) of the singularities (the justification of this technique can be found in [34] by Dumortier). After the first step of the blow-up procedure, system (7) can be written in the local map (2;, r/), where r / = y / z , as dz --at = ~x + Z2;~ - ~

d--/= ~ +

-

+ 1-r 2;

r162

x~),

r

and in the local map (0, y), where 0 = x/y, as

dt

dy !r d--7 = Zy + (~ + 1)Oy + Y

(12) y).

On the x = 0 axis of system (11) there is a singular point (0,0) which, by virtue of (8), is elementary (a node for c~ > 0 and a saddle for a < 0). On the y = 0 axis of system (12) there are no singular points, and therefore none of the trajectories of system (7) enters the origin in the direction of x = 0. Thus, after the first step of the blow-up all singular points proved to be hyperbolic (and, consequently, elementary). We can see from (11) and (12) that the perturbing terms r and r do not affect the scheme of elementary points of the singularity (for the terminology, see [28]). We can obviously use a polar blow-up instead of the a-process. We have used the a-process since it requires simpler calculations although it is easier to interpret the polar blow-up geometrically. The phase patterns of the singularity (0, 0) of the system dx d'--[

= a x 2 + fl 2;y - u s '

dy d--[

=

(a + 1)xy + 13y2

(13)

that satisfies conditions (8) are shown in Fig. 2.1, 1 (a > 0) and Fig. 2.1, 2 (a < 0). Their polar blow-ups are shown in the same figure under the numbers 6 and 7 respectively. Using the standard results of the theory of dynamical systems (see [24], for instance), it is easy to establish the existence of the homeomorphism h, which is close to an identity homeomorphism and which maps the trajectories of the polar blow-up of system (7) into the trajectories of the polar blow-up of system (t3). Note that the homeomorphism h is defined in 2124

0 Lf~

rex /

t"-,I

CO,

) 2125

the neighborhood of the glued-in circle and satisfies the relation h ( - x , - y ) = - h ( x , y). It follows that system (7) is locally topologically adjoined to system (13) by m e a n s of the homeornorphism that satisfies the a n t i s y m m e t r y property. Blowing up the singularity at the origin of system (9) with conditions (10), we rewrite (9) in the local maps (z, 7/) and (0, y) as follows:

d!= dt az

+ (/3 - 1)zr/+ l r

- - = - , +

xr/),

X

-

dt

dO --o + 02 - ~ d--/=

r

+

r

dy d-7 = & +

(15) - 1)@ + 1 ( @ , y ) Y

System (14) has two singular points, M~(0,0) and M2(0, 1), on the z = 0 axis. I a M1 the eigenvalues of system (14) are ,ka = a, ,k2 = - 1 and in M2 they are ,k~ = a + f l - 1, ,k~ = 1. By virtue of condition (10), the points M~ and M2 are elementary. System (15) has a singular point Ma(0, 0) lying on the y = 0 axis. M3 is an e l e m e n t a r y singularity since ,~1 = - 1 , ,k2 = / 3 r 0 at this point by virtue of (10). Thus the scheme of elementary points of the singularity of system (9) with condition (10) is defined by the 2-jet. As before, using the methods of [24] (the ,~-lemma, in particular), we can prove t h a t in the neighborhood of the origin the phase patterns of system (9) with condition (10) coincide with t h e phase patterns of the system

dx d--t = az~ + (/3 - 1)xy,

dy (a - 1)xy +/3y2. d'-~ =

These patterns are shown in Fig. 2.1, 3 (for a / 3 ( a + [ 3 - 1) > 0), Fig. 2.1, 4 (for a / 3 ( a + / 3 - 1) < 0, (a - 1)(/3 - 1)(a +/3) > 0), and Fig. 2.1, 5 (for a/3(a +/3 - 1) < 0, (a - 1)(/3 - 1)(a +/3) < 0). It is easy to verify that a/3(a +/3 - 1) < 0 implies (a - 1)(/3 - 1)(c~ +/3) -fi 0. "The polar blow-ups are shown in Fig. 2.1, 8, 9, and 10 respectively. We have the following important theorem as a corollary. T h e o r e m 2.1. The phase patterns of C2-typical fields of line elements in the plane in the neighborhood of a nontrivial singular point are ezhausted by the configurations shown in Fig. 2.1, 1 I - 1 3 and 15. Moreover, they and only they are C2-structurally stable. R e m a r k 2.1. The phase patterns shown in Fig. 2.1, I and 4 (11 and i4') in the neighborhood of (0,0) are topologically (but not C 1) equivalent. R e m a r k 2.2. The structural stability of the phase patterns 1-5 also follows from t h e results of Takens [56, pp. 68-69], who studied the germs of vector fields with a zero l-jet.

2.3.

N o r m a l F o r m s a n d B i f u r c a t i o n s of Fields of L i n e E l e m e n t s

If we regard system (5) as a formal power series with respect to the variables x and y (where r and r consist of monomials of even degrees N >_ 4), the following natural question arises: to what simplest form can system (5) be reduced by means of formal changes of coordinates and time? The importance of this question was noted by many authors (see, for instance, the works of Bogdanov [15] and Michel [49]), and the interest it attracts is partially due to the fact that it leads to the investigation 2126

of (typical) bifurcations in one-parameter families of fields of line elements. In the case of even degrees, this problem was investigated by Bronstein and Nikolaev [17]. They prove that in a typical (nonresonance) case the formal orbital normal form of system (5) contains one functional module (i.e., one unremovable monomial of degree N for every even N in decomposition (5)). The following statement is v~lid [17]. T h e o r e m 2.2. The formal vector field (5) of degeneration codimension < 2 caTz be reduced to one of the following formal orbital normal forms (the subscript shows the degeneration codimension). 1. Nonresonance normal forms:

wO'(x,y) = [am2+ (/3-1)mY]o-~ + [ ( a - l ) x y + 13y2+ ~ C"`y,,,,] 0

O--Y{;

.,=2

c~(a - 1)[2c~(1 - n ) - 111213(1 - k) - 1 1 [ 1 ( 2 l - 3 ) a + 1 - 211 + 113- 111 r 0

w(~

0 2]-~y 0 = [az~+13zY-y2+ L C.,y2.,] -O-~z+[(a+l)zy+13y

( V n , k , l > 2);

(a[2a(n-1)-1][Ic~+ll+1131] r 0 (Vn >2));

.,=2

w'1(z,y)= [a2+my+m=.. ~ C"xY'.,-'] O~zz+ [ ( a - l ) m y + y ' ] - ~ y O

(a[2a(l-n)-l]r

(Vn>_2))

m-~ 2

2. Resonance normal forms of codimension 1: 1

0:"~

ifa = 2(i - ,~)'

L D " ` z 2mO

ifa=O,

"`=2

m----9

O:~

g13

{ Dz2 ~O col~>(,,y) = w~o2~+

oo

oy CI: mO

ifa-

20 - k~;

=

1

2(,~- 1)'

ifa=O.

m=2

3. Resonance normal forms of codimension 2 : ~ C2kO Y -~z + Dz~n

1 1 ifa - 2(I - n------~'/3 -- 2(1 - k)' (?2C"`2m+Dy2k) 0 1 = ~z ifa=O, / 3 - 2 ( 1 - k ) ' Cxy2t_ 1 0 ira - 21- 1 o-S

2l~'

I]

""`=2

"~

e;(m, y) = w;

+

13 = i;

if a + l = f i - - O ;

C~"~

if ,~ - 2(1 -

~ D"`x2m

if a = O.

n)'

"`--=2

2127

The normal forms obtained make it possible to investigate local bifurcations in typical one-parameter and two-parameter families of vector fields (5). What is remarkable here is that bifurcations in such families do not lead to the appearance of topologically new phase patterns different from those shown in Fig. 2.1, I-5 (see [171). T h e o r e m 2.3. The phase patterns of systems with normal form d;? ) ( ~ ) ) are topologically equivalent to the phase patterns 3-5 (1 and 2). In the case of the normal form w~ (w~) we obtain the phase patterns topologically equivalent to ~ (5). Finally, in the case ofw+(w~) we arrive at the phase patterns 1 (2).

2128

Chapter 3 THE MORSE-SMALE

3.1.

FOLIATIONS

Globalization

Suppose that M denotes a 2-dimensional smooth connected compact manifold without boundary, a: M LM is a field of line elements on M, and t5' = {Xl,... ,zm} is a set of nontrivial singular points of the field a (m is an even number). We can assume without loss of generality that the restriction of the field a to every continuous closed curve, which does not pass through the singularity and is not homotopic to zero, defines two mutually opposite vector fields (we can always attain this by passing to a certain unramified double covering of the manifold M). According to the results obtained by Hurwitz, Shepardson, and Ezell (see [35]), there exists a closed surface M which is a double covering for M with points z l , . . . , xm as the ramification points for the covering M. Recall the corresponding construction. We fix a certain point z E M \ B and denote by g the genus of the surface M. Let r = 29 . We denote by a l , . . . , a , a set of closed curves which meet only at the point x and give a complete section of the surface M. Suppose, furthermore, that b l , . . . , bm is a set of pairwise nonintersecting arcs that connect the point x with x l , . . . , am respectively. Cutting M along the closed curves a l , . . . , ar and along the segments b l , . . . , b,,, we obtain a polygon Z with boundary bib'1.., bmb~ala2a'la~.., a,_lara;_la; (the asterisk denotes a change of orientation). Supose that Z is one more copy of the polygon Z. We denote its boundary by blb~.-./~mb~ala2a~a~.., ar_,a~a;_la;. In order to obtain a covering surface M, we have to glue the polygons Z and Z along their boundaries as follows: the side bl is identified (with due account of the orientation) with b~, b~ is glued to b~ (i = 1 , . . . , m), a~ is glued to a'k, and ak to a~ (k = 1 , . . . , r). According to the Riemann-Hurwitz formula, the genus ~0 of the surface M is 9=2g-1+--.

m

2

We denote by F the fundamental group ;rl(M) of the manifold M. Similarly, let F = ;rl(M). We can represent the manifolds M and ~-7 as M = A / F and M = A / F . Here A = {z e C, Izl < 1} is a universal covering, and F and P are completely discontinuous subgroups of the group of linear-fractional transformations. Since M is a ramified double covering of the manifold M, F is a subgroup with index 2 of the group F. Consequently, is a normal subgroup of the group F. Let g E F / F , and then g~ E F, i.e., J = g E E 7 / F and J is an involution of the space M. The inverse images y l , . . . , y m of the points z ~ , . . . , z ~ for the covering map ~r:M ~ M (and only they) are invariant under the action of the involution J. If x E M \ B , then ;r-~(x) consists of two different points, i.e., ;r-~(x) is a nontrivial orbit of the group F / F ~ Z2. We can show that for this field of line elements a there exists a vector field ~ -- (~:M---~ T M such that

DJ(~ o J) = - ~ ,

~,(z) = T~(~(~-'(z))),

(z r M \ B ) .

(16)

2129

3.2.

T h e o r e m on S t r u c t u r a l S t a b i l i t y

Let F*(M, B, d) (r > 2) denote the space of C~-smooth vector fields (: M- ---, TM- that satisfy condition

(16). L e m m a 3.1. There ezists an open and everywhere dense subset Fro in F~(M, 13, J ) such that every vector field in Fro has only a finite number of singular points. Proof. Recall that B = {z~,..., z,,} is a set of ramification points (i.e., nontrivial singularities). According to Lemma 2.2, we can assume that the singular points zi C B are of the types described by the lemma. Let U(z~) be small neighborhoods of z~. Repeating the arguments used in [24] for vector fields on M \ Up__~U(z~) and employing Field's Z2-equivariant theory of transversality [36], we get the statement of the lemma. Suppose that p E M is a singular point of the field of line elements e and U is a neighborhood of p. D e f i n i t i o n 3.1. The connection component of the intersection of the fiber of the foliation .T'~, defined by the field cr, with the neighborhood Up of the singular point p is known as a local fiber. D e f i n i t i o n 3.2. We say that the fiber l tends to the critical elements pi (i = 1, 2) if [\l = pl U P2 (the case pl = p~ is admissible). D e f i n i t i o n 3.3. The local separatrix for the singular point p is the local fiber 1~ which tends to p and at least on one side of l~ in the vicinity of p, there are local fibers which do not tend to the point p. The fiber l which contains the local separatrix l~ of the point p is called a (global) separatrix of the point p. The following proposition is a generalization of the Andronov-Pontryagin theorem to the case of fields of line elements. -

-

m

T h e o r e m 3.1. In the space r ' ( M ) --- P ( M , B, J) of all Cr-smooth vector fields ~:-M ~ T--M that satisfy condition (16) there ezists an open and everywhere dense subspace Fro such that the field of line elements a corresponding to the vector field ~ E F~o is structurally stable and satisfies the followiug conditions: (1) the field a has only a finite number of trivial singular points and limit cycles and all of them are hyperbolic; (2) the set of nontrivial singular points coincides with B and contains only singularities of the four types described in Theorem 2.1; (3) for the foliation which is defined by the field ~r every fiber, different from the critical elements, tends to a certain critical element," (4) there is no fiber l which would tend to the singular points p and q and be such that 1 is a separatriz for p and q at the same time (the case p = q is not excluded). The proof of the theorem consists of two parts. A. O p e n n e s s a n d S t r u c t u r a l Stability. We denote by A = { z l , . . . , x k } the set of trivial critical elements of the vector field ( (5 P;(M). Let B = {Yl,..-,Ym} be a set of nontrivial singular points. In the sequel, we consider a manifold M" which results from the manifold M upon a polar blow-up of all singular points of the set B. The natural projection p: M" ~ M is a diffeomorphism everywhere, except for the points lying on the circles {SI(B)},%I glued into M. The union Ui"=lSi(B ) of all these circles is the boundary OM* of the manifold M ' . It consists of integral trajectories of the vector field ( E F[~(M'), as is shown in Fig. 2.1, 6-10. R e m a r k 3.1. Instead of the manifold M" with the boundary OM" we can consider the manifold M'" obtained from two copies of the manifold M" as a result of the gluing along the circles SI(B). The manifold M'" has no boundary' and tile corresponding vector field ~ E l'~(M") of the manifold M'" contains only hyperbolic critical elements. 2130

Let us now define stable and unstable manifolds of elements that constitute the sets A and B. Since z E A is a hyperbolic critical element, the stable W'(z) and unstable W"(z) manifolds are defined as usual. After the use of the blow-up procedure, the point !/ E B is decomposed into hyperbolic equilibrium states, saddles and nodes. The nodal points lying on OM" are called half-nodes. The set of integral trajectories which enter a half-node as t ~ +oc (t --, - c o ) is called a 2-dimensional stable (unstable) manifold of the half-node. We understand a saddle element of a nontrivial singular point (unconnected, generally speaking) to be a subset of the circle of the polar blow-up which consists of all saddle points and, possibly, integral trajectories (provided that they exist) that connect saddle points. It is easy to see (see Fig. 2.1, 7-10) that in our case there exist four types of saddle elements. For instance, in Fig. 2.1, I0 the saddle element consists of two arcs .'-. ab and -"- cd of the glued circle. Every trajectory which does not lie on the circle of the polar blow-up and tends to one of the saddle points of this saddle element as t ~ +oo (t ~ - o o ) is said to be a stable (unstable, resp.) separatrix of the saddle element. We understand a stable (unstable) manifold of a saddle element to be the union of all its stable (unstable) separatrices. Thus, on every circle of a polar blow-up we distinguish a finite number of half-nodes and at most one saddle element. In what follows, we understand nontrivial critical elements to be half-nodes and saddle elements. In accordance with what was said above, there are eight types of critical elements, namely, nodes, half-nodes, closed orbits, saddles, and saddle elements of four kinds: (1) a circle containing two saddle points and two arcs connecting them, (2) a circle containing six saddle points and six arcs, (3) two isolated saddle points, and (4) two nonintersecting arcs with their ends at saddle points. D e f i n i t i o n 3.4. For the given vector field ~ E F ; ( M ' ) we define the phase p a t t e r n D as the set of its critical elements (trivial and nontrivial) with the following partial order: for a l , a2 E D we write al _< a2 if W"(aa) fl W'(a2) r e (in other words, there exists a trajectory going from aa to a2). Clearly, by virtue of condition (4) of the theorem and the fact that none of the pairs of sinks (sources) can be connected by a trajectory of the field ( E F ; ( M ' ) , the order _< is partial (i.e., is not defined for certain pairs of critical elements). Therefore, the phase diagram of the vector field ~r decomposes into three levels: (1) unstable elements (nodes, half-nodes, and closed orbits), (2) saddles and saddle elements, (3) stable elements (nodes, half-nodes, and closed orbits). Thus, the order is not defined for elements lying at the same level. D e f i n i t i o n 3.5.

Suppose that ( , ~ E I'~(M') and D , / ) are their phase diagrams. We say that D is isomor-

phic t o / ) if there exists a one-to-one correspondence h: D -~ /) such that (1) a E D is a critical element if and only if h(a) E /) is a critical element of the same type; (2) i f a l , a 2 E D, then a l _< a~ if and only if h ( a l ) _< h(a~). Our immediate aim is to show that after small perturbations of the vector field ~ E F~(M*) we have fields of the same class with isomorphic phase diagrams. In this case the main technical instrument is the concept of filtration. D e f i n i t i o n 3.6. Let ~ E F~(M'). The filtration connected with ~ is the sequence M0 C M1 C . . . C Mk = M* of compact submanifolds Mi of the manifold M" such that (1) for every i the boundary OMi can be represented as a disjunct union of smooth curves, each of which lies entirely in OM" or in M ' ; (2) the vector field ~ is transversal to OMi\OM" and ~,(Mi) C int Mi, t > 0; (3) the maximal ~cinvariant subset of the set M,+~\M, consists of a single trivial or nontrivial critical element c~i+~. In other words, ~tet~t(M,+l\intM,) = a,+~. Lemma

3.2.

Let ~ E F~(M'). Then there ezists a filtration connected with {. 2131

C2

Sq

Cq

C6 C4

S2 g~ '

C5

Fig. 3.1 Proof. Let a l , a 2 , . . . , a j be sinks (i.e., stable nodes, half-nodes, closed orbits) of the vector field {. We choose pairwise nonintersecting neighborhoods 1/1, V2,..., Vj of these critical elements with boundaries transversal to the field { everywhere, except for the points of oqM" (see Fig. 3.2). We set M~ = V~,M= = MI U V 2 , . . . , M s = a.i_l U ~~. Let us also suppose that c ~ j + l , . . . , a , is the set of all saddle elements and saddles of the field {. We shall first consider the stable separatrices of the saddle element c~j+l (i.e., the connected components of the manifold WS(aj+l\aj+~)) and construct, in the neighborhood U of the critical element aj+~, arbitrary sections S,, which are transversal to the rath separatrix of the saddle element aj+l. If U is sufficiently small, then the trajectories passing through the endpoints of the section oe,, cut OM transversally. Using the flow tubes of the field {, we can choose, in the vicinity of the arcs of the trajectories, certain transversal curves Co (n = 2m) which connect the endpoints S,, with cOk,l as is shown in Fig. 3.2 (where only nontrivial singular points with saddle elements are depicted). Let us now suppose that Vj is a domain bounded by the curves S,,,,Cn and the segment OMj. Let Mi+~ = Mj tO ~+1. It is easy to verify that Vh~n{,(~5+~) = aj+~ and Mj+~ satisfies t h e conditions of filtration. Using the same arguments in the case of other saddles and saddle elements, we arrive a t the chain of manifolds o = M o C . . . C Ms. Suppose, finally, that a s + l , . . . , (~k are the sources (i.e., unstable nodes, half-nodes, and closed orbits) of the field sr and let V , + l , . . . , '~ be their pairwise nonintersecting neighborhoods with boundaries transversal to { everywhere, except for the points of OM'. We set M,+~ = a l \ ( i n t Vs+2 tO... tOint Vk), Ms+2 = M \ ( i n t Vs+a U ... U int V k ) , . . . , M~ = M ' . It is easy' to see that ~ = Mo C MI C ... C Mk = .~,I" is the required filtration.

2132

02

% /

$2

I \ ~

O-I

S1

C4

C6 s3

~

c5

\1./

c4

S, I Fig. 3.2 The following proposition establishes the fact that the set F ; ( M ' ) is open in the space F ' ( M ") (in fact, this is a consequence of the fact that the submanifold OoMi = OMi\OM" is transversal to the vector field 9 F;(M')). L e m m a 3.3. We set ~ E F ; ( M ' ) . Then there exists a neighborhood U of the field ~ such that if ~ E U, then C E P[~(m') and the phase diagram of the field ~ is isomorphic to the field diagram of the field ~. We shall now prove tile structural stability of vector fields from I';(M'). To be more precise, we must establish the homeomorphism h, close to the identical one, which maps the trajectories of the perturbed field ~" E U into the trajectories of the unperturbed field (. P a r t 1. Let us first suppose that the vector field ~ does not have closed orbits. We shall consider the sink of the field ( and the corresponding sink c~(~') -= h(~), where r E U and h is the isomorphism of the phase diagram D and /) established earlier. Suppose that V is a disk containing the sink a, with V belonging to the stable manifold of the sink ~ and having boundary OV transversal to all ~ E U. Moreover, let c~(~') C W'(c~). We assume that cq, c ~ , . . , are saddles and saddle elements of the field ~r such that al _< ~ and let pl,p2,.., be points at which the unstable separatrices W~'(ai) cut OV. Similarly, we construct the points px((),P2((),.-, for the field (. Recall that the vector fields ~ and ( satisfy the condition of antisymmetry (16) with respect to the smooth involution J defined on the manifold M ' . According to (16), the following remark holds true. R e m a r k 3.2. Let ~ E F[~(M'). The involution J maps the sinks of the field ~ into the sources of the field and vice versa. The involution J maps every stable separatrix of the saddle s (the saddle element a) onto the unstable separatrix of the saddle J(s) (the saddle element ~). 2133

Let us now consider tile source J(c~) and assume that 81,82 . . . . is a set of saddles and saddle elements of the field ~ such that 8~ > d(o). We denote by ql,q2,.., the points at which the stable separatrices W,(Bi) of the elements 8r cut the boundary J ( a V ) of the source Y(c~). Note that qi = J(pi). We define the topological equivalence h between the vector fields ~ and r for" which purpose we first establish the homeomorphism between the sets aV t3 {qi} constructed for the vector fields ( and (. We set h(c~) = c~((), h(cq) = c~i((), h(pi) = pi((), h(qi) = JhJ-l(qi). We extend the homeomorphism h to the sets OV\{p~} and 0V\{pi(()} in a continuous arbitrary way, requiring, however, that the extension be uniformly continuous relative to the whole family of vector fields ( E U. We do the same to all other sinks of the field ~ (recall that they are finite in number). Thus, the homeomorphism h is defined on the sets 0V LI {qi}. Every point of the manifold M ' \ a M " (except for the equilibrium positions) attains the boundary aV of a certain sink or passes through one of the points qi. Let us extend the homeomorphism h to M ' \ a M " by means of the relation h~tz = (thz, where z ~ Ov u

{q~}.

It easy to see that h is invertible. It remains to prove that the mapping h is continuous. Our proof coincides in m a n y respects with that in [24]. The continuity of h is obvious in the sinks, sources, and in the neighborhoods of saddles and saddle elements. We shall prove the continuity of h at the points of stable separatrices of saddle elements. We shall consider, for definiteness, the polar blow-up of a nontrivial singular point of the "thorn" type (the other cases are similar). Let x E We(T), where T is a saddle element of the thorn type (see Fig. 3.3). We shall consider the sequence x,~ ~ z. We want to show that hz,~ ~ hz. Note that, by virtue of the continuity of the involution J, it suffices to prove that h is continuous at the point Jz E W"(T). We assume that S and ,9 are line segments which are transversal to ~ and pass through the points q and of the circle of the polar blow-up as shown in Fig. 3.3. Next, let Ii be the arc of the boundary 0V/of the sink a such that p~ E I~. Acting by the flow ~, on S, S, and Ii, we get tubular families for W~'(m) and W~(m), where m is a saddle point lying on the circle of the polar blow-up. The sets ~,(S), ~,(S), and ~,(I~) are fibers of the tubular families for l,V~'(m) and W ' ( m ) respectively. It is easy to verify that the mapping h maps the fibers of the tubular family of the flow ~L into the fibers of the tubular family constructed in a similar way for the flow if,. By virtue of the )~-lemma in [24], the fibers ~,(S), ~,(5"), and ~,(li) approximate the manifolds We(m) and W"(rn) in the CX-topology (as t ---, +cxz and t ~ - ~ respectively). The consideration of the projections 7r, and z~ of the points J(z,~) onto W ' ( m ) and W"(m) proves the continuity of the mapping h ([24]). P a r t 2. If the vector field ~ C F[~(M') has closed orbits, then the conjugation construction is somewhat more difficult. However, since closed hyperbolic trajectories are trivial critical elements, our proof coincides with that in [24]. We have thus proved that the field ~ C F~o(M") is structurally stable. Remark

3.3.

The homeomorphism h constructed above commutes with the involution J.

Indeed, as was shown above, the homeomorphism h is first established between the sets OV U {qi}, on which h commutes with J. Since the flows ~, and ~ satisfy the antisymmetry condition (16) with respect to the involution J , we immediately see that h commutes with the involution J by construction, i.e., h o J = Joh. Let us consider the field of line elements a: M ~ LM defined on the 2-manifold M. The field a leads to the foliation with singularities defined on M. Two foliations 9v and .T"~ are said to be homeomorphic (equivalent) if there exists a homeomorphism h: M ~ M, which maps the foliation .T- into the foliation Y'~, i.e., h maps the fiber that passe~ through p into the fiber that passes through h(p). R e m a r k 3.4. The foliations ~- and U' on the manifold M are homeomorphic if and only if there exists a homeomorphism h: M" ---* M ' , defined on the covering manifold M ' , which preserves the trajectories on M" and commutes with the involution J. 2134

V. 1

r ~

cl

q

Fig. 3.3 B. Density. The proof of the fact that the set of vector fields F ; ( M ' , B, J) with a fixed collection of nontrivial singular points B and involution J is everywhere dense in the space Fr(M ", B, J) repe-ts, in many respects, the proof for the Morse-Smale vector fields on two-dimensional orientable manifolds (see [24], for instance). In what follows, we use tile terminology from [24].

2135

We assume that M is an orientable compact 2-manifold. If ~ E F~(M ") has singular points (trivial or nontrivial), all of which are hyperbolic, and there exists a nontrivial recurrent trajectory, then the field ( can be approximated by the field ( which has one saddle connection more than (. We take the point p lying on a nontrivial recurrent trajectory and the neighborhood O v such that Op C/J(Ov) = o. We choose the required perturbation A~ such that A~ - 0 outside of Op U J(Op), namely, we define A( on Op as is done in [24] and extend it to J(Ov) by means of the involution J. In the same way we can eliminate all nontrivial recurrent trajectories 7~, J('~) at the expense of the appearance of new saddle connections s~, J(s~). Thus the field ~ which has nontrivial recurrent trajectories can be approximated by the field ( which has only trivial recurrent trajectories and a finite number of saddle connections si. Let us show how the saddle connection sl, not lying on the boundary OM*, can be destroyed in a finite number of steps by means of the perturbations of the field ('. We consider the saddle connection s. Reasoning as above, we construct a perturbation localized in a small neighborhood O~ of the point p E s and then extend it to J(Op). The end of the proof coincides exactly with that in [24]. We have thus completed the proof that P~o(M", B, J) is everywhere dense in the space F~(M", B, J). Consequently, we have proved the Andronov-Pontryagin theorem.

3.3.

T h e Topological Classification of t h e M o r s e - S m a l e F o l i a t i o n s

The Andronov-Pontryagin theorem characterizes structurally stable foliations on two-dimensional orientable manifolds and leaves open the questions about distinguishing between the classes of topological equivalence and the description of the totality of these classes. Below we consider a discrete object, the (generalized) Peixoto graph, which can be constructed from the foliation U and is a complete topological invariant of the foliation. Suppose that U is a structurally stable foliation on the compact surface M and ~ : M ~ T M is the corresponding vector field on the ramified double covering of the manifold M (see Sec. 3.1). The field satisfies the condition of antisymmetry (16) with respect to the involution J. Let A U B U C be a set of critical elements of the vector field ( which consists of rn trivial singular points a E A, n nontrivial singular points b E B, and p cycles c E C. R e m a r k 3.5. If the vector field ~ has p hyperbolic limit cycles, then it can always be replaced by the vector field ~"on the manifold N obtained from M by cutting it along each of the p cycles and subsequent contraction of the edges of the cut into a point. The field ~" has 2p nodal points more than the field ~, and the manifold N is, in general, unconnected. R e m a r k 3.6. If B is nonempty and contains nontrivial singularities shown in Fig. 2.1, 1 and 5, then, by using homotopy, we can replace the vector field ~" in the neighborhood of these singularities by the vector field 77, as is shown in Fig. 3.4, 1 and 2. After every operation of this kind, the field r/ will have two nodal points more than ~', and, as a result, its set B will consist only of the singularities shown in Fig. 2.1, 2 and 3. We can choose the fields ~ and 7] satisfying the antisymmetry condition with respect to certain involutions. Having realized the chain ~ ~ ~" ~ q, we introduce the following definition. D e f i n i t i o n 3.7. The scheme of the foliation ~" is a 3-level graph X, the set of whose vertices consists of singular points of the vector field 77 and the set of whose edges consists of the set of separatrices of this field. The upper level denoted by a and the lower level denoted by ~o are filled in by the sources and sinks of the field r] respectively; the middle level, which we denote by c~, is filled in by 2-, 4-, and 6-separatrix saddles. The structure of the incidence of the graph can be established as follows: the separatrix of the saddle is associated with the edge which connects the saddle with the source (sink) lying at the c~ (aJ)-level, the direction on the edge being the direction of the vector field 7] on the corresponding separatrix. In addition, 2136

Fig. 3.4 (1) on the graph X, mutually nonintersecting pairs of sources (sinks) are marked which were obtained from the unstable (stable) limit cycles upon passage from the vector field ~ to the vector field ~ (see Remark

3.5); (2) on the graph X, subgraphs are marked, each of which consists of three vertices and two edges appearing as the result of the homotopy of the vector field r into the vector field r/ (see Remark 3.6); (3) if the set B is nonempty, then the involution J is defined on the graph X which leaves 2- and 6separatrix saddles (and only them) fixed and maps 4-separatrix saddles into one another. The vertices of the a-level are mapped into the vertices of the w-level and vice versa, i.e., J(a) = ca, J(w) = a (see also Remark 3.2). It is shown in Peixoto's work [53] that even in the case of flows the scheme of a flow is not its complete topological invariant. A similar example for foliations is considered in [17]. The following definition is important for an understanding of the sequel. D e f i n i t i o n 3.8 (see [42]). The system of rotations R on the graph X is a cyclic order established on the set of edges adjoining every vertex of the graph. The graph X, together with the system of rotations R defined on it, is denoted by X R. If X is a scheme of the foliation ~-, then X R is known as the Peixoto graph of the foliation 5r . The main property of a system of rotation is that it allows us to realize the algorithm of finding faces. Suppose that X is a graph and R is its rotation system. We take an arbitrary vertex vl E X and an edge a~ adjoining vl. Suppose that v2 is a vertex of the graph X connected with vl by the edge a~ and b~ is the edge of the vertex v2 lying in the cyclic order on the right of a ~ . Moving along the edge b~ to the vertex v3, we determine the edge c~3, which lies on the right of b~. Continuing the process by induction, we finish it on the edge z~, if two subsequent edges in the chain are again a~ 1 and b,,2. We thus isolate the cycle a ~ , b ~ , . . . , z ~ , of length n on the graph X which defines the face F1 on X. To find the face ?'2, we must proceed from the edge which lies on the right of any edge of the face F1 and is such that the angle between them was absent in Fl and use the construction indicated above. All faces Fl, F2,..., F~ on the graph X will be found when all the angles are used. 3.2. The structurally stable foliations .Tzl and .7z2 are topologically equivalent if and only if their Peizoto graphs are isomorphic, i.e., X th "~ X t~.

Theorem

The proof repeats that of the structural stability of foliations carried out in Sec. 3.2. The main here that the isomorphic Peixoto graphs define tile same faces of X when X is immersed in a surface same character of adjointness of the faces to one another (see algorithm of finding the faces). The the Peixoto graph are elementary cells of the vector fields rh and r/2 corresponding to the foliations

thing is and the faces of -~"l and 2137

cA

Q

%

c~

0 cO

2

1

3

t 4

5

Fig. 3.5 -7"2. Their similar adjointness allows us to extend the homeomorphism h that preserves the trajectories to the whole manifold N. Theorem 3.2 reduces the classification of structurally stable foliations to the problem of the classification of graphs provided with rotation systems. Let us now consider the converse question, namely, what condition must the rotation system ~ on the abstractly given scheme X satisfy for the foliation 5t" to exist (on a certain surface M) with X 7~ as a topological invariant?

D e f i n i t i o n 3.9. An abstract scheme is a 3-1evei orientable (unconnected, generally speaking) graph ,a:"with 2-, 4-, and 6-valent (power) vertices. We denote these sets by o'2, a4, and or6 and position them on the middle level, a. All the vertices of the a-level have the same number of edges entering a vertex and issuing from it. Every vertex of the c~-level is connected in some way with the vertices of the upper level (a-level) and the lower level (w-level) by means of the entering and, correspondingly, issuing edges. T h e Euler characteristic ]a t + I w l - la41 - 21a61 (where the module is the number of vertices of a given type) of every connected component of the graph is an even number belonging to the infinite sequence {2, 0 , - 2 , - 4 , . . . } . In addition, the following conditions are satisfied: (1) a set (possibly empty) of nonintersecting pairs of vertices is isolated on the set of vertices of the aand w-levels; (2) if 2( contains vertices of the type a2 and a6, then the involution d is defined on 2( which leaves the vertices of the types a2 and aG fixed and maps the vertices of the type a4 (provided t h a t they exist) into one another. The vertices of the c~-level are mapped by the involution d into those of the w-level, and vice versa; (3) 2-edge subgraphs a - s - b, where a E a, b E fl, are isolated for some vertices s E a2 a n d / o r s E a6, and these subgraphs are invariant under the involution J; 2138

(4) if the connected component Xpo C X of the graph X has no vertices on the a-level, then it consists of a single c~- and a single ~a-vertex connected by a single edge. D e f i n i t i o n 3.10. A standard molecule and Andronov's molecule are connected orientable subgraphs of the graph X shown in Fig. 3.5, 1 and 2; see also [53, p. 394]. They are associated with elementary cells shown in Fig. 3.5, o~ and 4D e f i n i t i o n 3.11. The abstract scheme X is known as Peixoto graph X x: if there exists in X a rotation system 77. such that each of the cycles that isolate the faces F 1 , . . . , Fm on the graph X is isomorphic either to a standard molecule or to Andronov's molecule. Theorem

3.3.

The Peizoto abstract graph X 7r is realized as the Peizoto graph X n of a certain foliation 5c.

Proof. According to the Heffter-Edmuns theorem (see [42] and also [43] and [25, p. 240]), every rotation system 77. on X defines a single cell immersion of X into the surface M. By the hypothesis, the faces of this immersion are either standard cells or Andronov's cells (no other kinds of cells are possible on orientable surfaces by virtue of the structu:'al Mayer theorem [21]; see also [1]). Since an orientation was defined on X, it remains to fill in the cells obtained in a standard way and get the orbits of the vector field 7/. The inverse chain of transformations r/-+ ~" --+ ~ leads to the required foliation .T.

2139

Chapter

TOPOLOGICAL

CLASSIFICATION

4

OF

QUASIMINIMAL

FOLIATIONS

CONTAINING

SETS

Introduction In this chapter, We consider the problems of topological classification of foliations containing quasiminimal sets on the assumption that the foliations have singularities and are defined on closed orientable surfaces M of genus p, p >_ O. The topological key to all the results included in this chapter is the study of the arrangement of nonclosed fibers of foliations with saddle singularities of an integral and semi-integral index on these surfaces. The asymptotics of lifting these fibers to the universal covering (ramified or unramified) generates a topological invariant, namely, an orbit of a homotopical class of rotation that generalizes the classical Poincar6 number of rotation. For a deeper understanding of this chapter we refer the reader to the reviews by Aronson and Grines [7, 1] which discuss the problems of topological classification of flows (continuous-time dynamical systems) defined on closed surfaces. However, in contrast to flows, the consideration of foliations requires a more complicated technique since the foliations may be nonorientable, i.e., nonimmersible in a flow, and this requires the use of the ramified covering apparatus. The topological classification of foliations is usually understood as the solution of the following problems. (1) The finding of topological invariants which give the necessary and sufficient conditions for topological equivalence. The collection of these invariants is called a complete topological invariant of a foliation. (2) The finding of the whole set of admissible topological invariants and use of every admissible collection for constructing the invariant of a "standard representative" that defines the class of topologically equivalent foliations (realization problem).

4.1.

T h e H o m o t o p i c Class of R o t a t i o n . T h e O r b i t of a H o m o t o p i c Class o f R o t a t i o n

Unless otherwise specified, here and in what follows M means a closed orientable surface (a twodimensional manifold) of genus p, p _ 0. We denote by E(m) = { s ~ , . . . , s m }

(m is even) the set of points

marked on M , and if m = 0, then we set E (m) = o. We denote by Ao, A~, A2 the following classes of surfaces: 1. A0 is a two-dimensional sphere S 2 with four marked points. 2. A1 is a two-dimensional torus T 2 without marked points. 3. A2 is either a surface of genus p >_ 2 without marked points or a surface of genus p _> 1 with the number of marked points equal to 2 or exceeding 2, or o~ with the number of marked points equal to 6 or exceeding 6. In the sequel we shall consider only these classes of marked surfaces. According to the Klein-Poincar6 uniformization theorem, M can be represented as -M/G, where M is a circle x 2 + y 2 < 1, G is a discrete group of transformations on M-, and the covering ~r:M ~ M for m = 0 is universal nonramified, and in the case m # 0, universal ramified with a ramification of order two over the 2140

marked points E (m). group G (it is known (1) If M belongs ramifications of order

The circle E: x 2 + y2 = 1 is called the absolute. Here are the main properties of the as a covering group). to the class A0, then, since for S 2 the torus 7 `2 is a ramified double covering space with two over the four marked points, any element g E G has the form 1

1

x ' = v ~ ( ( - 1 ) ~ x +,/X~),

y'= ~ ( ( - 1 ) k y

+,/X,),

where k,r,s E Z (Z is a group of integers),

~x = 1 - x 2 - y2,

~ = zx + ((-1)~x + v ~ r ) ~ + ((-1)~y + v ~ s ) ~.

(2) If we have the class A1 (a two-dimensional torus without marked points), then every element g E G has the form x' 1

= ~(~

+ 4Xr),

y' = ---~(y + ,/Z~),

i

D = A + (z + vfAr): + (y + x/-As)2, i.e., for this element the whole absolute E is stationary. (3) If we have the class A~ and E (m) = o, then G is a finitely generated Fuchsian group consisting only of hyperbolic elements. Any nonidentity element g E G has exactly two fixed points (one attracting and the other repulsing) that lie on the absolute. (4) If we have the class A2 and E (m) -fi o, then G is a finitely generated Fuchsian group consisting only of hyperbolic elements and elliptic elements of order two. D e f i n i t i o n 4.1. The rational points of the absolute are its points which, for the classes of surfaces Ao, At, have the coordinates P q

4~+--~'

~ '

p, q e Z ,

and for the class of surfaces A: the points of the absolute which are fixed points of the hyperbolic elements of the group G, different from the identity element. The other points of the absolute are irrational. In the case m -# 0, besides the ramified universal covering ~r: M --, M we shall consider the ramified double covering q : / ~ --* M with ramifications of order two over the marked points E(m). In this case, two points lie on M over every unmarked point on M and exactly one point over every marked point. Moreover,

7r = qql, where ql: M ~ M is a nonramified universal covering. According to the Hurwitz formula, the genus /~ of the surface 251 is equal to 2p - 1 + m/2, and the closed orientable surface M can be represented as M/F, where F is a subgroup of index two of the group G. Since t5 is an integer, rn is even. In the case of a surface of the class A2, every hyperbolic element g 6 G that does not belong to the subgroup F possesses the property 92 6 F. In what follows, in addition to the properties of the group G, we shall need some properties of the automorphisms of the group G described in [4]. It is known [50] that any homeomorphism f : M --* M, f ( E ( " ) ) = E (m) is lifted to the homeomorphism f : M --* M" which leaves the set a'-~(E (~)) invariant and induces the automorphism r of the group G according to the law

,(a) = / g / - ' ,

g e c:

and conversely: any automorphism r: G --* G induces the homeomorphism fi:M- ---* M which leaves the set 7r-~(E (m)) invariant and blows down onto M by means of the mapping 7r. In addition, it is known [4, 52] that the automorphism r : G ~ G can be uniquely extended to the absolute with the aid of a certain 2141

homeomorphism r ' : E ---, E (in [521 this is proved for the case M 7{ S 2, E (m) = ~, in [4] for the case where a ramified universal covering is considered, for the case 5'2, 2:('~) # e inclusive). Let H be all automorphisms of the group G. It is known that any automorphism of the group G uniquely induces the homeomorphism of the absolute. \ge denote by H ~ the set of all homeornorphisms of this kind. We introduce the concept of a homotopic rotation class and the orbit of the hornotopic rotation class of the semifiber L of the foliation F on M. By Q~m) we denote the class of foliations on M such that if the foliation F r Q~'), then it has a nonempty finite set of singularities which are topological saddles of zero integral or semi-integral index, where m = 0 if F has no thorns, and if a thorn does exist, then m means the number of saddles of semi-integral index, i.e., the number of saddles with an odd number of separatrices, including the thorns (saddles with one separatrix). We say that the set of marked points of the foliation F E Q~'~) is the set of all its saddles with a semiintegral index if thorns exist, or an empty set if there are no thorns. We denote the set of marked points of the foliation F E QI m) by E]('~). Let us consider the semifiber L of the foliation F E QI m), which does not contain singularities, and denote by f : M ---* M any homeomorphism that maps the set of marked points ~(m) into t h e set of marked points E (m) and L0 = f(L). We define on L the parametrization t E [0,+oc) such that t = 0 corresponds to the initial point of the semifiber L and, as t varies from 0 to + ~ , runs through the whole semifiber L. Then, by virtue of f , the parametrization t E [0, +0r

is also induced on L0 =

f(L).

Let us a s s u m e that if ]~(m) = o or

if ~(m) = E(m), then f = id (an identity mapping). We denote by lo the connected component of the complete inverse image 7r-l(L0) on M with the induced (by virtue of 7r) parametrization and by or(10) the set of all limit points 10 lying on t h e absolute (a priori a(lo) can be either an empty set or a single point of the absolute, or a certain subset of points of the absolute). D e f i n i t i o n 4.2. The homotopic class of rotation of the semifiber L of the foliation F E Q~"~) is the set of points of the absolute ~(L) = U g(~(10)) 9 gEG

For flows on M (in this case the set of marked points 12(") on M is an empty set), the concept of the homotopic rotation class was introduced by Aranson and Grines in [9], and for foliations with singularities it was introduced by Aranson in [4, 5]. It should also be pointed out that for flows on M no constraints are imposed on t h e set and the topological types of equilibrium states when a homotopic rotation class is introduced since a nonrarnified universal covering is used. D e f i n i t i o n 4.3. The orbit O,~(L) of the homotopic rotation class of the semi fiber L of the foliation F E Q[") is the set of points of the absolute O~

=

H~

where H ~ is the set of homeomorphisms of the absolute induced by all automorphisrns of the covering group G. Note that since the set of all automorphisms of the group G is countable, the sct of all homomorphisms from H ~ is also countable. In addition, Ore(L) does not depend on the choice of the homeomorphism f on M such that f : ~(m) __~ E(m). We denote by Q2 the class of foliations on M such that if the foliation F E Qe, then every fiber of F contains not more than one singularity and the fibers which contain a singularity do not make loops. Suppose that L is a semifiber of the foliation F 6 Q~m) 71 Q2 that does not conta.in singularities, ~(m) is a set of marked points for F, and f: M ~ M is an5' homeomorphism that maps 9. ('~) into Z ('~). We again 2142

denote L0 = f(Lo); by 10 we denote the connected component of the complete inverse image rr-l(Lo) on and by a(lo) the set of all limit points l0 lying on the absolute. T h e o r e m 4.1 ([4]). direction on -M.

or(10) is only one point of the absolute. In this case we say that lo has an asymptotic

R e m a r k 4.1. If L is a semifiber of the foliation F E QI m) N Q2 that does not contain singularities, then there can only be the following possibilities for L: (1) either L is a nonclosed Poisson stable semifiber, (2) or L is a nonclosed Poisson unstable semifiber which has a nonclosed Poisson stable semifiber in its limit set, (3) either L is a closed semifiber, (4) or L is a nonclosed semifiber which has a closed semifiber in its limit set. Then in the situation described in Theorem 4.1 the point c~(10) of the absolute is an irrational point in cases (1) and (2) and a rational point in cases (3) and (4).

4.2.

N e c e s s a r y a n d Sufficient C o n d i t i o n s of Topological E q u i v a l e n c e o f T r a n s i t i v e Foliations

The transitive foliations on M, i.e., foliations containing everywhere dense semifibers, are a natural generalization of the irrational winding of the torus T 2, and therefore the classification of these foliations is of particular interest. It is known [29, 25] that for two transitive foliations F, F' without singularities on T 2 to be topologically equivalent, it is necessary and sufficient that their Poincard rotation numbers be countable by means of an integer-valued unimodular matrix. For transitive flows on M this problem was solved by Aranson and Grines in [9], and for transitive foliations with saddle singularities on M it was solved by Aranson [8, 9] (for S 2, see also the work [30] by Aranson and Zhuzhoma). Here are the results obtained in [4, 5, 30]. We denote by Qa the class of foliations on M such that if F E Qa, then F is a transitive foliation. T h e o r e m 4.2. For two foliations F, F' E (~Im) n 0,2 A Q3 to be topologically equivalent, it is necessary and sufficient that there exist singularity-free semifibers L, L' of the separatrices of the foliations F, F', which have similar orbits of the homotopic classes of rotation, i.e., Ore(L) = Om(L'). In the case of the class of surfaces A0 (a sphere S 2 with four marked points) and, correspondingly, transitive foliations on S = with four thorns, Theorem 4.2 admits of a simple formulation in the language of the rotation orbit offoliations [30] since in this case the orbit of the homotopic class of rotation of the semifiber does not depend on the choice of the semifiber. Remark

4.2.

There exists a continuum of topological classes of transitive foliations belonging to the class

of foliations Q~'~) fq Q= v1 Qa (for the same m).

4.3.

N e c e s s a r y a n d Sufficient C o n d i t i o n s of Topological E q u i v a l e n c e of S i n g u l a r i t y - P r e e Q u a s i m i n i m a l Sets of Foliations. T h e P r o b l e m of L a b y r i n t h s

Using the method of ramified coverings, we can solve the problem of topological equivalence not only of transitive foliations oil M but also the problem of topological equivalence of quasiminimal sets that are not necessarily transitive. To be more precise, this is a problem of immersion of quasiminimal sets into the 2143

manifold M. It is posed as follows: let f~, f~' be quasiminimal sets of foliations F, F' on M ( F and F ' may coincide). We have to find the necessary and sufficient conditions for the existence of a homeomorphism f: M ~ M such that f would map the fibers of tile quasiminimal set f~ into the fibers of the quasiminimal set f~'. If we find such a homeomorphism f , then we say that f] and f~' are topologically equivalent. By analogy with flows on M, where this problem was solved by Aranson and Grines for nontrivial minimal sets of flows on M of genus p _> 2 [10, 11] and also for nowhere dense nontrivial m i n i m a l sets of flows without an equilibrium state on the torus T 2 (see [14]), we shall consider for foliations only the case of quasiminimal sets that do not contain singularities (see [4]). This is the answer to one of the problems on labyrinths posed by Rosenberg [55]. Suppose that F E Q~m) f3 Q2 and f~ is its quasiminimal set. We say that fi satisfies condition A if (1) fl does not contain singularities of the foliation F ; (2) every connected component cr of the set M \ f t contains exactly one singularity of the foliation F. It can be immediately established that this f~ is nowhere dense on M. Fibers of the first kind of the set ~ are fibers that belongs to the boundaries, a t t a i n a b l e from within, of the connected components of the set M\f~. The other fibers from f~ are fibers of the second kind. 4.3 ([4]). Suppose that f~, ~2' are quasiminimal sets of the foliations F, F' e Q~m) O Q~ and ~, fl' satisfy condition A. Then, for f~, f2~ to be topologically equivalent, it is necessary and suO~cient that there ezist two semifibers of the first kind L, L' of the sets ~, f~' which have similar orbits of hornotopic rotation classes, i.e., Ore(L) = Om(L'). Theorem

Remark

4.3.

There exists a continuum of topological classes of quasiminimal sets of foliations from the

class Q~m) Cl Q2 (with the same m) such that these quasiminimal sets satisfy condition A. In [4], cases where there may be singularities of the foliation in f~ are also considered.

4.4.

R e a l i z a t i o n of T r a n s i t i v e F o l i a t i o n s a n d S i n g u l a r i t y - F r e e Q u a s i m i n i m a l

Sets of Foliations

Somewhat earlier in this chapter, we introduced a complete topological invariant (the orbit of a homotopical rotation class) for transitive foliations and quasiminimal sets of foliations such t h a t these quasiminimal sets contain no singularities. We have thus solved the first problem of the t,opological classification of objects of this kind. Let us now solve the second problem of topological classification, namely, the problem of admissibility and construction of a standard representative. If we take transitive foliations without singularities on the torus T 2 as an object, therh as we know, in this case tile complete topological invariant is the number of Poincar6 rotations co with accuracy to within the recount by means of the integer-valued unimodular matrix [29, 25]. In this case, co is an irrational number. Every irrational number can be the Poincar~ rotation number w for the class of transitive foliations without singularities on T 2, i.e., all irrational numbers are admissible, and using a n y co of this kind, we can construct a standard representative, namely, the irrational winding of the torus (foliation on T 2 whose fibers are nonclosed geodesics in the metric of zero curvature on the torus). The resulting foliation has the given number co as its rotation number. A complete topological invariant is also known for foliations without singularities on the torus T 2 which is not transitive and has no closed fibers. It is the so-called characteristic class [4, 14], for which the problem of admissibility and realization is also solved in [4]. In this section, we shall only solve the second problem of topological classification for transitive foliations and quasiminimal sets without singularities defined on M. For flows these problems were considered in [ll, 5], and for foliations in [4]. We must understand what irrational points of the absolute E: x 2 + y2 = 1 are admissible for these foliations and know how to construct "standard representatives" from the admissible points. 2144

'vVe shall consider the class of surfaces A2. In this case M either has genus p _> 2 and there are no marked points on M, or M has genus p >_ 1 and the number of marked points is greater than or equal to two, or M is a sphere 5'2 and the number of marked points on it is larger than or equal to six. In this case the surface M is represented as M / G , where G is a finitely generated Fuchsian group all of whose nonidentical elements are either hyperbolic (if there are no marked points) or hyperbolic and elliptic of order two (if there are marked points). We denote

Q= 0 Ore, Om=Olm nQ2; m----O

then @,~ gl @m, : o if m r m', and, therefore, if the foliation F E Q, then the n u m b e r m and the class | are uniquely defined for F . Let, for definiteness, m r O. Then, considering on M the smooth metric without singularities d of constant curvature k : - 1 such that the square of the element of the arc has the form ds ~ = 4(1 - x ~ - ~,~)-~(d2 + dU~), we find that d is invariant with respect to the group G, and then, under the projection ~r: M + M , the metric d of constant curvature k = - 1 is induced on M everywhere except for the marked points g (m) E M , where the metric d degenerates. Therefore, the geodesics on M- in the metric d (they are the arcs of the circles orthogonal to the absolute) are omitted under the mapping z into the geodesics on M in the metric d. We have also represented M in the form M = M / F , where F is a subgroup with index two of the group G, ~'1 is a closed orientable surface of genus /5 = 2p - 1 + rn/2 which is a ramified double covering with ramifications of order two over the points Z (m) E M.

The natural projections were denoted by ql, q, 7r,

~r = qqx. Let us denote by A1, A2, A3, A4 the following sets: Al is a set of geodesics on M" in the metric d such that for any geodesics l E Al there exists an element "t E F such that 7(l) = l, and whatever element /3 E G we take, the geodesic fl(l) either coincides with l (being a set) or has no points in common with l; A2 is a set of geodesics on M resulting from the following operation: we take various topological limits of geodesics from A~ and delete those of the topological limits which are points of the absolute; A3 is a subset of the set A2 such that (1) any curve L = 7r(l), l E A3, is a nonclosed curve, (2) any ray L0 C L is Poisson stable; A4 is a subset of the set A3 such that every connected component of the set M \ f l o is simply connected, where flo is the closure of the curve L C M, L = 7r(l), l E A4. We introduce the following notation: DI(E) is a set of boundary points, lying on the absolute E: z 2 + y2 = 1, of geodesics l E A3. We say that the set D~(E) is a set of admissible points on the absolute E for quasiminimal sets of foliations M without

singularities; T ~ = E x E is a two-dimensional torus on which a Euclidean metric is defined as a metric of the direct product E x E. The torus T 2 = E x E is known as an absolute torus; D~(T 2) is a set of admissible points on an absolute torus for singularity-free quasiminimal sets of the foliations on M such that they are distinguished by geodesics l E A3 and every pair of b o u n d a r y points el, e2 of the geodesic 1 is regarded as one point (ex, e2) on the absolute torus T 2 = E x E; Do(E) is a set of boundary points, lying on the absolute E, of the geodesics l E A4. T h e set Do(E) is said to be an admissible set on the absolute E for transitive foliations on M; Do(T 2) is an admissible set on the absolute torus T 2 for transitive foliations on M. It is distinguished by the geodesics from A4. The pair of boundary points of every geodesic of this kind is regarded as one point on the absolute torus 7'". 2145

The properties of these admissible sets will be refined in the sequel. Since L = ql(/), l E At, is a closed curve on ~-I, which has no serf-intersections in a single traverse along i,~ and Z, is nonhomotopic to zero on .~.'I, the problem of finding the set A1 is algorithmically solvable [54]. But then it is meaningful to speak of the sets A2, Aa,.44. We shall describe the problem of realization and admissibility of the sets D I ( E ) , D I ( T 2) for quasiminimal sets f~ which satisfy condition A introduced above, such that ~2 belongs to the foliations F of the class Q. R e a l i z a t i o n . We fix on the absolute any pair of points el,e2 which are boundary points of the geodesic l (E Aa and denote by (e~,e2) E D l ( T 2) the corresponding point on the absolute torus E x E. Then, since l (E Aa, it follows that L = 7r(l) is a nonclosed Poisson stable geodesic in the metric d on M and /, has no self-intersections. We denote G'(I) = Ug~a9(/) and by fl we denote the union of all topological limits, lying on M, of the sequences of geodesics from (7(I) and f~ = 7r(~). It is proved in [4, Lemma 13] that [2 is a nowhere dense geodesic distribution in the metric d on M consisting of nonintersecting geodesics without self-intersections. Now it is easy to fiber M - fl into curves so that a foliation F results on M whose ~ is a quasiminimal set satisfying condition A. A d m i s s i b i l i t y . Suppose that a foliation F E 0~ is given on M and fl is its quasiminimal set satisfying condition A. It is proved in [4, Lemma 15] that in that case the orbit of the homotopic class of rotation O~(/,) of any semifiber L C f~ belongs to DI(E). Moreover, if we take any f i b e r / J C fl (by condition A it does not contain singularities of the foliation F) and denote by I~ its connected inverse image on M" and fix its lirrfit points e~,e2 on the absolute E, then the point (e~,e2) and the point (e2, e~) belong to the set D~(T2). The proof of this is based el, e2 E E obtained from l', we the homeomorphism homotopic of fl and f]0 in the situation of

on the fact that if we connect on M by geodesics all these pairs of points get a geodesic distribution [2o, which is topologically equivalent by means of to the identical quasiminimal set f~ (we understand the topological equivalence Theorem 4.3).

It turns out [4, L e m m a 14 and the remark connected with it] that the sets D~(E), D~(T 2) have the following properties: (1) the set D~(E) is everywhere dense on the absolute E, has a power of continuum and a zero Lebesgue measure; (2) the set D~ (T 2) is nowhere dense on the absolute torus T 2 = E x E, has a power of continuum, is a set of Lebesgue measure zero, and does not contain isolated points. Carrying out the factoring operation with the use of certain sets, Aranson [4, Theorem 13 and its corollary] also solved the problem of realization and admissibility of the sets D0(E), D0(T 2) for transitive foliations F of the class Q =

~ ~ where ~)m = QIml n Q~ n pa. Um=OQI, fact that Do(E) C DI(E), Do(T 2) C D~(T2), the

Despite the sets Do(E), Do(T 2) have the following properties: (1) the set Do(E) is everywhere dense on the absolute E, has a power of continuum and a Lebesgue measure zero; (2) the set Do(T 2) is nowhere dense on the absolute torus T ~ = E x E, has a power of continuum, is a set of Lebesgue measure zero, and does not contain isolated points. Note that in [4, 11] the same language is used to solve the problem of realization and admissibility of nowhere dense nontrivial (different from the equilibrium state and closed trajectories) minimal sets of flows and transitive foliations on closed orientable surfaces. The results given in Chap. 4 can be generalized to closed nonorientable surfaces M of any genus ]5 _> 1, but in this case we must first lift the foliations 2~, defined on /~, into the foliations F defined on the double (nonramified) covering M which is a closed orientable surface of genus p = /5 - 1. However, additional difficulties are encountered here and, to our knowledge, this work has not yet been completed, although there are some results (see, for instance, [12]). This work was supported by the Russian Fund for Fundamental Research (grant 93-01-01407) and the International Research Fund (long-term grants by tile International Science Foundation, 99300 and RZA000).

2 |46

REFERENCES 1. D. V. Anosov, S. Kh. Aranson, 1. Yu. Bronstein, and V. Z. Grines, "Smooth dynamical systems-l," Contemporary Problems of Mathematics, Fundamental Trends, Vol. 1, ltogi Nauki i Tekhniki, Moscow, 151-242 (1985). 2. D. V. Anosov, S. Kh. Aranson, V. Z. Grines, P. V. Plykin, E. A. Sataev, A. V. Safonov, V. V. Solodov, A. N. Starkov, A. M. Stepin, and S. V. Shlyachkov, "Dynamical systems with hyperbolic behavior," Contemporary Problems of Mathematics. Fundamental Trends, 66. ltogi Nauki i Tekhniki, 5-247 (1991). 3. S. Kh. Aranson, "Trajectories on nonorientable two-dimensional manifolds," Mat. Sb., 80 (122), No. 3, 314-333 (1969). 4. S. Kh. Aranson, Topological Classification of Foliations with Singularities and Homeomorphisms with lnvariant Foliations on Closed Surfaces. Part 1. Foliations [in Russian], Lobachevsky Gorky State Univ. Press, Gorky (1988). 5. S. Kh. Aranson, "On the topological equivalence of foliations with singularities and homeomorphisms with invariant foliations on two-dimensional manifolds," Usp. Mat. Nauk, 41, No. 3 (243), 167-168

(1986). 6. S. Kh. Aranson, "On some arithmetical properties of dynamical systems on two-dimensionM manifolds," Dokl. Akad. Nauk SSSR, 222, No. 2, 265-268 (1975). 7. S. Kh. Aranson and V. Z. Grines, "Topological classification of flows on closed two-dimensional manifolds," Usp. Mat. Nauk, 41, No. 1 (247), 149-169 (1986). 8. S. Kh. Aranson and V. Z. Grines, "Topological classification of cascades on closed two-dimensional manifolds," Usp. Mat. Nauk, 45, No. 1 (271), 3-32 (1990). 9. S. Kh. Aranson and V. Z. Grines, "On some invariant dynamical systems on two-dimensional manifolds (the necessary and sufficient conditions of topological equivalence of transitive systems)," Mat. Sb., 90 (132), 372-402 (1973). 10. S. Kh. Aranson and V. Z. Grines, "On topological invariant minimal sets of dynamical systems on two-dimensional manifolds," in: Qualitative Methods of the Theory of Differential Equations and Their Applications. Sci. Notes of Gorky Stage Univ., E. A. Leontovich-Andronova (ed.), Gorky State Univ. Press, Gorky, No. 187 (1973), pp. 3-28. 11. S. Kh. Aranson and V. Z. Grines, "On the representation of the minimal sets of flows on two-dimensional manifold by geodesics," Izv. Akad. Nauk SSSR, 42, No. 1, 104-129 (1973). 12. S. Kh. Aranson and E. V. Zhuzhoma, "On the structure of quasiminimal sets of foliations on surfaces," Mat. Sb., 185, 31-62 (1994). 13. S. Kh. Aranson and E. V. Zhuzhorna, "On the trajectories of covering flows in the case of ramified coverings of a sphere and a projective plane," Mat. Zametki, 53, No. 5, 3-13 (1993). 14. S. Kh. Aranson and E. V. Zhuzhoma, "On the topological equivalence of nowhere dense minimal sets of dynamical systems on a torus," lzv. Vuzov. Mat., 5 (168), 104-107 (1976). 15. R. I. Bogdanov, "Local orbital normal forms of vector fields in the plane," Tr. Petrovsky Sere., No. 5, 51-84 (1979). 16. I. U. 13ronstein and 1. V. Nikolaev, "Structurally stable singular points of the field of line elements in the plane," Izv. Akad. Nauk Moldova. Mat., 3, 23-29 (1992). 2147

17. I. U. Bronstein and I. V. Nikolaev, "Orbital normal forms and bifurcations of the field of line elements in tile plane," Diff. Uravn. (in press). 18. A. A. Kadyrov, "Singular points of differential equations with nonorientable trajectories in the plane," Diff. Uravn., 19, No. 12, 2039-2048 (1984). 19. I. P. Kornfeld, Ya. G. Sinai, and S. V. Fomin, Ergodic Theory [in Russian], Moscow (1980). 20. L. D. Landau and E. M. Lifshits, Theoretical Physics, Vol. 7, The Elasticity Theory [in Russian], Moscow (1987). 21. A. G. Mayer, "On trajectories on orientable surfaces," Mat. Sb., 12, No. 1, 71-84 (1943). 22. V. V. Nemytsky and V. V. Stepanov, Qualitative Theory of Differential Equations [in Russian], Moscow (1949). 23. P. V. Plykin, "Sources and sinks of A-diffeomorphisms," Mat. Sb., 94, No. 2, 243-264 (1974). 24. J. Palls and W. de /vlelo, Geometric Theory of Dynamical Systems. An Introduction. Springer, New York (1982). 25. L. E. Reisins, "Topological classification of dynamical systems without rest points on a torus," Latv. Math. Yearly, 5, 113-121 (1969). 26. G. Ringel, Map Color Theorem, Springer, Heidelberg (1974). 27. A. S. Sonin, Introduction to Liquid Crystal Theory [in Russian], Nauka, Moscow (1983). 28. K. S. Sibirsky, Introduction to the Algebraic Theory of the lnvariants of Differential Equations [in Russian], Schtiintza, Kishinev (1983). 29. S. V. Fomin, "On dynamical systems with a purely point spectrum," Dokl. Akad. Nauk SSSR, 77, No. 1, 22-32 (1951). 30. S. Kh. Aranson and E. V. Zhuzhoma, "Classification of transitive foliations on the sphere with four singularities of spine type," Selecta Math. Sov., 9, No. 2, 117-121 (1990). 31. I. U. Bronstein and I. V. Nikolaev, "Structurally stable fields of line elements on surfaces," Nonlinear Analysis (submitted). 32. I. U. Bronstein and h V. Nikolaev, "On Peixoto graphs of Morse-Smale foliations," CRM-2214, Univ. de Montrdal, preprint, 1-23 (1994). 33. T. Cherry, "Topological properties of solutions of ordinary differential equations," Amer. J. Math., 59, 957-982 (1937). 34. F. Dumortier, "Singularities of vector fields on the plane," g. Diff. Eq., 23, 53-106 (1977). 35. C. L. Ezell, "Branch point structure of covering maps onto nonorientable surfaces," Trans. Amer. Math. Soc., 243 123-133 (1978). 36. M. Field, "Une version d,~t~ivariant de thdor/eme de la transversalitd de Thorn," C. R. Acad. Sci. Paris, 282, 379-381 (1976). 37. V. Guinez, "Positive quadratic differential forms and foliations with singularities on surfaces," Trans. Amer. Math. Soc., 309, No. 2, 477-502 (1988). 2148

38. V. Guinez, "Nonorientable polynomial foliations on tile plane," J. Diff. Eq., 87, No. 2, 391-411 (1990). 39. C. Gutidrrez, "Structural stability for line fields with singularities on two-dimensional manifolds," Lect. Notes Math., 486, 17-19 (1975). 40. C. Gutidrrez and J. Sotomayor, "An approximation theorem for immersions with stable configurations of lines of principal curvature," Lect. Notes Math., 1007, 332-368 (1983). 41. C. Gutidrrez and J. Sotomayor, "Structurally stable configurations of lines of principal curvature," Asterisque, 98-99, 195-215 (1982). 42. J. L. Gross and T. W. Tucker, Topological Graph Theory, Wiley, New York (1987). 43. L. Heffter, "Uber das Problem der Nachbargebiete." Math. Annalen, 38, 477-508 (1891). 44. G. Levitt, "Feuilletages des surfaces," Ann. Inst. Fourier, 32, 179-217 (1982). 45. G. Levitt, Feuilletages des Surfaces. These de doctorat, Univ. Paris (1983)i 46. G. Levitt, "Pantalons et feuilletages des surfaces," Topology, 21, 9-33 (1982). 47. G. Levitt and H. Rosenberg, "Differentiability and topology of labyrinths in the disk and annulus;" Topology, 26, No. 2, 173-186 (1987). 48. N. Markley, "On the number of recurrent orbit closures," Proc. AMS, 25, No. 2, 413-416 (1970). 49. M.-F. Michel, "Formes normales de certaines equations differentieles quadratiques," C.R. Acad Sci., Paris (to appear). ,50. I. V. Nikolaev, "Foliations with tripod and thorn singularities on the torus and sphere," CRM-2196, Univ. de Montrdal, Preprint (1994). 51. I. V. Nikolaev, "Foliations with singularities of semi-integer index," CRM-2206, Univ. de Montrdal. Preprint (1994). 52. d. Nielsen, "Uber topologische Abbildungen geschlossener Flachen," Abh. Math. Seminar, Hamburg Univ., 3, No. 1,246-260 (1924). 53. M. M. Peixoto, "On the classification of flows on 2-manifolds," Dynamical Systems. Proc. of Syrup. held at Univ. of Bahia, Brazil Acad. Press, Brazil (1971), pp. 389-419. ,54. B. L. Reinhard, "Algorithms for Jordan curves on compact surfaces," Ann. Math., 75, No. 2, 209-222 (1962). 55. H. Rosenberg, "Labyrinths in the disk and surfaces," Ann. Math., 117, No. 1, 1-33 (1983). 56. F. Takens, "Singularities of vector fields," Publ. IHES, 43, 47-100 (1973). 57. W. P. Thurston, The Geometry and Topology of a-Manifolds. Mimeographed Lecture Notes, Princeton Univ., Chaps. 1-9 (1978). 58. A. V. Zorich, "Hamiltonian flows on multivalued Hamiltonians on closed orientable surfaces," Preprint. MP1-94-29, Max-Planck Institut fur Mathematik, Bonn.

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