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Lecture Notes in Mathematics Edited by A. Dold and 6.Eckmann Subseries: Department of Mathematics, University of Maryland Adviser: M. Zedek
Anatole Katok Jean-Marie Strelcyn with the collaboration of F. Ledrappier and F. Przytycki
Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities
Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo
Authors
Anatole Katok Mathematics 253-37, California Institute of Technology Pasadena, CA 91125, USA Jean-Marie Strelcyn Universite Paris-Nord, Centre Scientifique et Polytechnique D6partement de Math~matiques Avenue J.-B. CI6ment, 93430 Villetaneuse, France Fran£ois Ledrappier Laboratoire de Probabilit6s, Universit6 Paris VI 4 Place Jussieu, ?5230 Paris, France Feliks Przytycki Mathematical Institute of the Polish Academy of Sciences ul. Sniadeckich 8, 00-950 Warsaw, Poland
Mathematics Subject Classification (1980): Primary: 28 D 20, 34 F 05, 58 F 11,58 F 15 Secondary: 34C35, 58F08, 58F 18, 58F20, 58F22, 58F25 ISBN 3-540-17190-8 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-1 ? 190-8 Springer-Verlag New York Berlin Heidelberg
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specificallythose of translation, reprinting, re-useof illustrations,broadcasting, reproduction by photocopyingmachineor similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "VerwertungsgesellschaftWort", Munich. © Springer-Vertag Berlin Heidelberg 1986 Printed in Germany Printing and binding: DruckhausBeltz, Hemsbach/Bergstr. 214613140-543210
TABLE OF CONTENTS Introduction PART
I.
E X I S T E N C E OF INVARIANT WITH S I N G U L A R I T I E S
MANIFOLDS
(by A. KATOK
STRELCYN)
and J.-M.
i.
Class
2.
Preliminaries
3.
Overcoming
4.
The Proof
5.
The F o r m u l a t i o n of Pesin's Manifold Theorem
6.
Invariant
(1.1) 7.
PART
of T r a n s f o r m a t i o n s
-
of Lemma
10
Topics
Abstract
19
Invariant 24
for Maps
Satisfying
Conditions
(1.3)
25
ABSOLUTE
Properties
of Local
Stable 35
CONTINUITY
41
and J.-M.
STRELCYN)
i.
Introduction
2.
Preliminary
3.
Some Facts
4.
F o r m u l a t i o n of the A b s o l u t e a Sketch of the Proof
5.
Start of the Proof
6
The F i r s t M a i n Lemma
41 Remarks
and N o t a t i o n s
from M e a s u r e
Theory
42
and Linear A l g e b r a
Continuity
Theorem
55 62 65
7
Start of the Proof Projection
and C o v e r i n g
9
Comparison
of the V o l u m e s
- II
79 Lemmas
i0
The Proof of the A b s o l u t e
ii
Absolute
Continuity
12
Infinite
Dimensional
13.
Final
88 107
Continuity
of C o n d i t i o n a l
Theorem
Measures
Case
(by F. L E D R A P P I E R Introduction Preliminaries
3.
Construction
4.
Computation
130
154
THE E S T I M A T I O N OF E N T R O P Y FROM B E L O W T H R O U G H LYAPUNOV CHARACTERISTIC EXPONENTS
2.
117
138
Remarks
i.
46
and
- I
8
IIIo
I
of S i n g u l a r i t i e s
3.3 and Related
Manifolds
(by A. K A T O K
PART
Singularities
5
Influence
Some A d d i t i o n a l Manifolds II.
with
F O R SMOOTH MAPS
and J.-M.
and F o r m u l a t i o n
157
STRELCYN)
of the Results
157 162
of the P a r t i t i o n of E n t r o p y
167 175
IV
PART
IV.
THE E S T I M A T I O N OF E N T R O P Y FROM ABOVE LYAPUNOV CHARACTERISTIC EXPONENTS (by A. K A T O K and J.-M.
1
Introduction
2
Preliminaries
3
Construction
4
The Good and Bad E l e m e n t s
of P a r t i t i o n s
The Main Lemma The E s t i m a t i o n
PART V.
Introduction
2.
Terminology
193
199
SYSTEMS
199 and N o t a t i o n
The M a p p i n g
5.
The A p p l i c a b i l i t y Ergodic T h e o r e m
Billiards. ¢.
200
Generalities
The C o m p u t a t i o n
Set.
201
of
of the O s e l e d e c
d#
207
Multiplicative 222
6.
The S i n g u l a r
7.
The B i l l i a r d s of Class ~ . lld¢II and lld2~ll
8.
Proof of Lemma ations
7.4.
P r o o f of Lemma Inequality
7.4.
REFERENCES
DYNAMICAL
196
STRELCYN)
The Plane
2.
184 189
3.
APPENDIX
~t
of E n t r o p y
4.
Final
183
{~t}t~l of P a r t i t i o n
PLANE B I L L I A R D S AS SMOOTH WITH SINGULARITIES
i.
10.
I BO
E S T I M A T I O N OF E N T R O P Y OF SKEW P R O D U C T F R O M ABOVE T H R O U G H V E R T I C A L L Y A P U N O V C H A R A C T E R ISTIC E X P O N E N T S
(by J.-M.
9.
of the R e s u l t
181
6
i.
180
STRELCYN)
and F o r m u l a t i o n
5
APPENDIX
THROUGH
The B i l l i a r d s
of Class
229
P
The rate of G r o w t h
Part One:
Elementary
237
Configur249
Part Two:
P ro o f of the Main 258 273
Remarks OSELEDEC
of
MULTIPLICATIVE
ERGODIC
THEOREM
276 279
i.
INTRODUCTION During the past t w e n t y - f i v e years the h y p e r b o l i c p r o p e r t i e s of
smooth dynamical systems
(i.e. of d i f f e o m o r p h i s m s and flows) were
studied in the ergodic theory of such systems in a more and more general framework [Rue]2,3).
(see
[AnO]l,2,
[Sma],
[Nit],
[Bri],
[Kat] I,
[PeS]l, 3,
The d e t a i l e d h i s t o r i c a l survey of the h y p e r b o l i c i t y and
its role in the ergodic theory up to 1967 is given in
[Ano]2, Chapter ].
One of the most important features of smooth dynamical
systems
showing b e h a v i o r of h y p e r b o l i c type is the e x i s t e n c e of invariant families of stable and unstable m a n i f o l d s and their so called "absolute continuity".
The m o s t general theorem c o n c e r n i n g the
e x i s t e n c e and the absolute c o n t i n u i t y of such families has been proved by Ya. B. Pesin
([PeS]l,2).
The final results of this theory give a partial d e s c r i p t i o n of the ergodic properties of a smooth dynamical an a b s o l u t e l y continuous
system w i t h respect to
invariant m e a s u r e in terms of the L y a p u n o v
c h a r a c t e r i s t i c exponents.
One of the m o s t striking of the many
important consequences of these results d e s c r i b e d in
[pes]
is the 2,3 so called Pesin entropy formula which expresses the entropy of a smooth d y n a m i c a l system through its L y a p u n o v c h a r a c t e r i s t i c exponents. Our first m a i n purpose is to g e n e r a l i z e Pesin's results to a
broad class of d y n a m i c a l systems with s i n g u l a r i t i e s and at the same time to fill gaps and correct errors
in Pesin's proof of absolute
c o n t i n u i t y of families of invariant m a n i f o l d s
([Pes] I, Sec.
3).
We
followed Pesin's scheme very closely and this may at least partly e x p l a i n the length of our p r e s e n t a t i o n and heaviness of details, e s p e c i a l l y in Part II. (and unstable)
Parts I and II c o n t a i n the theory of stable
invariant m a n i f o l d s
c o r r e s p o n d to the context of
in our more general s i t u a t i o n and
[Pes] I.
At the end of Part II we also
prove an infinite d i m e n s i o n a l c o u n t e r p a r t of Pesin's results [Pes] i" The m o t i v a t i o n
for our g e n e r a l i z a t i o n lies in the fact that some
important d y n a m i c a l systems o c c u r r i n g in classical m e c h a n i c s example,
from
(for
the m o t i o n of the system of rigid balls w i t h elastic
collisions)
do have singularities.
the example mentioned) Briefly speaking,
Some of these systems
(including
can be reduced to s o - c a l l e d b i l l i a r d systems.
a b i l l i a r d system describes the m o t i o n of a point
mass w i t h i n a R i e m a n n i a n m a n i f o l d w i t h b o u n d a r y w i t h r e f l e c t i o n from the boundary. Our general c o n d i t i o n s on the s i n g u l a r i t i e s f o r m u l a t e d in Sec. 1 of Part I g r e w out of an attempt to u n d e r s t a n d the nature of s i n g u l a r i t i e s
in the b i l l i a r d problem.
VI
Since
a Poincare
flow u s u a l l y
has
singularities
In Part
whose
(first-return
may also provide
and c o n t i n u o u s
formula.
map
singularities,
time d y n a m i c a l
essential
III we prove the b e l o w
to
in smooth
case
tion
from below.
of Parts
line.
It seems
with
of d i s c r e t e
for the Pesin
changes
time
This
I and
Recently
II.
ingenious
completely simpler
that Mane's
entropy
the paper
[Pes]2, 3.
very
avoids
and it is s u b s t a n t i a l l y
Sinai-Pesin
estimate
an a l t e r n a t i v e
His proof
treatment
with minor
[Sin] 1 and
way the results
gave
manifolds
a unified
for a smooth
transformations
systems.
This part r e p r o d u c e s
idea goes back
map on a section)
considering
[Led]
i' in an
proof uses R. Mane
proof
([Man]l)
of the estima-
the use of i n v a r i a n t
than the p r o o f
method
along
can be applied
the to our
case. The above e n t r o p y
estimate
proved
in Part
IV is largely
independent
of the rest of the book. In
[Pes]
P e s i n derives from his results on invariant stable and 3 m a n i f o l d s the d e s c r i p t i o n of ergodic p r o p e r t i e s of a smooth
unstable dynamical
system
on the i n v a r i a n t
All his a r g u m e n t s property
with
literally
Bernoulli Jacobian
property
apply
from
the g r o w t h
of p e r i o d i c satisfies
points
Sec.
1 of Part
continuous class convex with
invariant measure
includes
the extra
finite
all c o m p a c t
and concave
IV the P e s i n
not know w h e t h e r nents
holds
Let us notice proof
that
exponents
arcs
every
C3
for s o - c a l l e d
of such
of the
is s a t i s f i e d
systems.
M. W o j t k o w s k i
Sinai-Bunimovich
through
measure
This
of
line intervals,
of Parts
of III and
for such billiards.
estimate
invariant
from
absolutely
convex arc has the t a n g e n c y By the results
that
I.
by a finite number
and s t r a i g h t
and
assuming
to the natural
class
bounded
the above e n t r o p y
recently
II. entropy
the s i n g u l a r i t i e s
with respect
of class
formula
between
1 of Part
of
of the
in Part
in our s i t u a t i o n
from Sec.
for a b r o a d
for an a r b i t r a r y that
estimate
and show that the c o n d i t i o n s
all its tangents.
entropy
stronger
the c o n n e c t i o n
regions
assumption:
order w i t h
that the proof
in great detail
I are s a t i s f i e d
exponents.
of B e r n o u l l i
It seems
also hold
for plane b i l l i a r d s
Lyapunov
of his proof
than the one o b t a i n e d
the c o n d i t i o n s
In Part V we study map
a somewhat
map
[Kat] 2 c o n c e r n i n g
the m e a s u r e
Poincare
to our case.
requires
of the P o i n c a r ~
Results
set: w i t h n o n - z e r o
the sole e x c e p t i o n
We do
the L y a p u n o v
expo-
for such a billiard.
([WOJ]l, 2) found billiards
an easy
the L y a p u n o v
are non-zero.
Resuming,
one can say that
lower r i g h t corner
in the p r e s e n t
of the f o l l o w i n g
diagram,
book we c o m p l e t e d
the
VII
The theory of A n o s o v systems
The theory of b i l l i a r d s of
and of the r e l a t e d systems
Sinai and B u n i m o v i c h
as A x i o m A systems,
etc.
i
I
Pesin Theory of m a p p i n g s w i t h
Pesin Theory of diffeo-
> singularities
m o r p h i s m s of compact manifolds
A concise resume of the m a i n results of the p r e s e n t book can be found in
[Str].
Other p r e s e n t a t i o n s of Pesin's theorem c o n c e r n i n g the e x i s t e n c e of i n v a r i a n t m a n i f o l d s were given later by D. Ruelle A. Fathi, M. Herman and J.-C. Yoccoz
([Fat]).
several g e n e r a l i z a t i o n s of that theorem a class of i n f i n i t e - d i m e n s i o n a l maps
([Rue] 1 ) and
D. Ruelle has d e v e l o p e d
(non-invertible smooth maps,
([Rue]2,3)).
R. Ma~e has found
another i n f i n i t e - d i m e n s i o n a l v e r s i o n of Pesin's t h e o r e m The authors w o u l d like to point out in the p r e p a r a t i o n of this book. w r i t t e n by the second author.
([Man]2).
their unequal p a r t i c i p a t i o n
A l m o s t all the text was a c t u a l l y
The first author suggested the general
plan of the w o r k and w o r k e d out the arguments w h i c h allow us to overcome the p r e s e n c e of s i n g u l a r i t i e s
in the c o n s t r u c t i o n of i n v a r i a n t
m a n i f o l d s and in the above entropy estimate.
Naturally, we d i s c u s s e d
together numerous q u e s t i o n s c o n c e r n i n g p r a c t i c a l l y all subjects treated in the text. The first draft of the theory d e s c r i b e d in the p r e s e n t book was p r e s e n t e d by the second author in D e c e m b e r 1978 at the Seminar of M a t h e m a t i c a l Physics at IHES
(Bures-sur Yvette, France).
The m a t e r i a l
of this book r e p r e s e n t s a part of the "Th~se d' Etat" of the second author, d e f e n d e d 30 April 1982 at U n i v e r s i t y Paris VI
(France).
Our n o t a t i o n s are very similar to those used by Pesin, but they are not the same. C o n c e r n i n g the e n u m e r a t i o n of formulas,
theorems,
etc, the first
number indicates the section in which the given formula, is contained.
The lower Roman numeral
In the interior of the same parts, Despite all our efforts,
theorem,
etc.,
indicates the part of the book.
the Roman numerals are not marke~.
some m i s t a k e s can remain.
g r a t e f u l to the readers kind enough to point them out.
We will be
Viii
Acknowledgments.
This book owes very m u c h to Dr. F. L e d r a p p i e r
(CNRS, U n i v e r s i t y Paris VI, France)
and to Dr. F. P r z y t y c k i
(Mathemati-
cal Institute of Polish A c a d e m y of Sciences, Warsaw). Besides being a c o - a u t h o r of Part III, F. L e d r a p p i e r made numerous useful remarks c o n c e r n i n g other topics treated in the book.
In partic-
ular he played a very i m p o r t a n t role in the e l a b o r a t i o n of the infinite d i m e n s i o n a l case. The role of F. Przytycki can hardly be overestimated.
We owe him
the final f o r m u l a t i o n of conditions c h a r a c t e r i z i n g our class of maps w i t h singularities.
In the previous v e r s i o n s conditions on the growth
of the first d e r i v a t i v e as well as of the growth of the two first d e r i v a t i v e s of the inverse m a p p i n g near the s i n g u l a r i t i e s w e r e assumed. Using ideas of F. P r z y t y c k i we were able to dispose of these conditions in Parts I-III and c o n s e q u e n t l y to extend the class of m a p p i n g s under consideration.
We thank sincerely both of them.
We also thank Dr. G. B e n e t t i n M. Brin
(University of Maryland,
nique, Palaiseau, Poland),
France),
Dr. Ya. B. Pesin
of Dijon, France), France)
(University of Padova, USA),
Dr. P. Collet
Dr. M. M i s i u r e w i c z (Moscow, URSS),
Dr. J.-P. T h o u v e n o t
and Dr. L.-S. Young
very useful discussions.
Italy), Dr.
(Ecole P o l y t e c h -
(University of Warsaw,
Dr. R. R o u s s a r i e
(University
(CNRS, U n i v e r s i t y Paris VI,
(Michigan State University,
U.S°A)
for
In p a r t i c u l a r the first author d i s c u s s e d the
early v e r s i o n of the theory d e s c r i b e d in this book w i t h Ya. B. Pesin who made several useful remarks. i m p o r t a n t formula
(4.10) v.
G. B e n e t t i n c o m m u n i c a t e d to us
M. M i s i u r e w i c z
found the c o u n t e r e x a m p l e
d e s c r i b e d in Sec. 7.8 v. We thank also Dr. R. D o u a d y Dr. M. Levi
(Boston University,
of Wroclaw,
Poland)
(Ecole P o l y t e c h n i q u e , U.S.A.)
Palaiseau,
and Dr. T. N a d z i e j a
France),
(University
for their help in the final editing of the text.
We would e s p e c i a l l y like to a c k n o w l e d g e the advice and gentle c r i t i c i s m of Dr. R. Douady,
whose careful reading of the m a n u s c r i p t
enabled us to m a k e m a n y c o r r e c t i o n s and improvements. It is our p l e a s a n t o b l i g a t i o n to express our g r a t i t u d e to the D e p a r t m e n t of M a t h e m a t i c s of the U n i v e r s i t y of M a r y l a n d and the N a t i o n a l Science F o u n d a t i o n for the support given to the second author for his trips to the U.S.A.
(NSF Grant MCS79-030116).
The
second author also thanks the D e p a r t m e n t of M a t h e m a t i c s of the Centre S c i e n t i f i q u e et P o l y t e c h n i q u e of U n i v e r s i t y Paris XIII, e x c e l l e n t w o r k i n g conditions.
for
PART I EXISTENCE
OF I N V A R I A N T
MANIFOLDS
A. K a t o k
1.
CLASS
i.i
OF T R A N S F O R M A T I O N S
In S e c t i o n
formulate
basic
consideration moderately Let following (A) of c l a s s
is "thin"
conditions contains
There
such that
we denote
b)
(TyV, II'IIy)
respectively,
every
Obviously compact
We w i l l equal
to
m
s(e)
satisfying
an o p e n s m o o t h
the
(at l e a s t m.
C < i, R < l, g >_ 1
x
and
w h e r e by
to the set
of the e x p o n e n t i a l
such t h a t
w = exPxly
p(x,y)
and
q
p(x,X)
X c M, and by map
eXPx:
< Rv(X),
linear maps ÷
T x V + V.
one has
lld(exPx I) (Y)II < q.
n o r m of the
(or s h o r t l y
Here
d eXPx(W) : (TxV,II'IIx)
(TxV, ll'iIx),
N'II) d e n o t e s
the n o r m in
TxV
metric.
e > 0
where
condition
there p(x,y)
exists
w = exPxl(y) and
say that the m e t r i c
V
re > 0
such that for
< m i n ( r E , R v ( X ) ) def R E ( x )
(B) is a l w a y s
manifold
and
satisfied
o n e has
lld(exPxl(y)]l _C(min(R, [0(x,M\V) ]g)) def Rv(X )
lld eXPx(W)II 0
and
d > 0
= T' nx(A n(x)).
B n (x)
holds. 3.3•
Let us define
B n(x) = T'nxeXp nx(V n(x)
is a neighborhood of the origin in IRm. Suppose that x ( A k n ~ ,y. We can define a map ~' : nx
B
n
÷ IRm
(x)
in the following way -i -i ~'nx = T'@n+l x o exp n+ 1 x o ~ o exp ~nx o (T' @nx)
(3.13)
It follows from (3.3.1) that ~' is a diffeomorphic embedding nx ! o~ Bn(X) into ~m. Moreover, t h e l i n e a r p a r t o f ~nx at the origin preserves the decomposition ]Rm = ]Rk ~ IRm-k Thus, we can represent
~'n x
in a "coordinate form": (3.14)
~nx(U,V) = (AnxU + anx(U,V),BnxV + bnx(U,V)) where Anx
u c ]Rk , v 6 ]Rm-k, #n+l x = T' o (d~ nXlEl~nx)
Bnx = T'Cn+ix o (d~ n x IE
o
( ~n x) T' -i,,1
) o (T'nx) 2@nx
a
nx and bnx are C P mappings da nx (0,0) = 0, dbnx(0,0) = 0. Since the maps
anx(0,0)
= 0, bbx(0,0)
! Ty, y ( A, are isometries,
= 0,
it follows from (2.2.3)
that
IIAnxll < I' (x)
(llBnZxlI) -I
1
> J
(3.15)
17
Let
tnx =
(anx,bnx).
The f o l l o w i n g
theorem
dure of the r e d u c t i o n manifold
3.1.
There
x E A k N Qt~,y, iIdtnx(Zl)
~emark.
n ~ 0, Zl,Z 2 E B n (x)
b
2.2,
Here
appears
Proof.
for the
into
and the
first
in a c r u c i a l
abstract
proce-
invariant
= {w E T nxN;
The map T n + i x N.
(2.8),
IiZl _ z2II,
A(x)
and
~(x)
come
from
respectively.
the fact that
#
has h i g h e r
smooth-
way.
Ilwll n x < (e(x)~2blnl) 2
An(X) ~
nx
is a c o n v e x
is d e f i n e d
We have
for
neighborhood
on
A (x) n
and t r a n s f o r m s
(z I) - d ~ n x
' (z 2) - d~nx' (0)) II = IId #nx
= Iid~nx (w I) - d ~ n x
of the o r i g i n
in
this
set
Zl,Z 2 E Bn(X)
IIdtnx(Zl ) - d t n x ( Z 2) II = II(d~nx ' - (d~nx
for all
one has
functions
time
such that
stated,
and c o n s e q u e n t l y , N.
H > 0
< yb~(x)HA(x) [~bu(x)]n
and f o r m u l a s
As a l r e a d y
An(X)
~n x
in the s u b s e q u e n t
to P e s i n ' s
a constant
_ dtnx(Z2)ii
(]..3) T h e o r e m
T
exists
the c o n s t a n t
ness
the key role
situation
theorem
THEOREM
where
plays
of our
(Zl)
-
d
(0)) ' Cnx
+ (3.16)
(z 2) II =
(w 2) If', J
where
wi =
(T'nx)-l(z i)
E T nxN,
i = 1,2,
and
II~II' means
the o p e r -
I
ator n o r m for the linear ~:
,
(T ~nxN' II'II~n x
N o w we can use the norms From
operator
II" IIx (2.2.5)
.
+
(2.2.5)
generated and
,
(T n+ixN, II II n+l x and
(2.2.1)
)
to relate
by the R i e m a n n i a n
(2.2.1),
it is easy
the norms
metric
to see that
p.
!
If"IIx
with
18
l!~II' ~ blnl _ ~ 1 (t
> Q(x)~2bln]
because
from
into origin implies
(3.11)
I
c6 yb b - ~
The prooof
f
be a
C2
one has
of
Q(x) ~nx
(3.3.5)
mapping
space
and every
E
Thus
~
Mn(X)
contains
An+l(X).
subset
Euclidean
This
An+l(X).
is based on the following
from an open convex
maps origin
nx
lies outside
(~Mn(X))
V
space.
easy inequality. of
0
of
Then for every
one has ( sup O~s~l
IId2f(su) H)IluIIIlhN.
from the obvious
= d f ( O ) (h)
4bln ]
(Q(x) ~2blnl) 2 1
b Y 2tc2(~) b"
into another
lldf(u) (h)H ~ lldf(0)(h)ll-
Indeed,
Q2(x)
c6Q(x) Q(x) e 2blnl b ) > 2t ¥
and the image
some Euclidean u 6 V
2bln
that the image of the ball
(3.3.5) Let
Q(x)
+
(4.9)
identity
(df(su) (h))ds 0
and from From such that
~s(df(su) (h)) = d2f(su) (u,h) . (3.3.4)
one has
~n~(An+l(X)) c Mn(X ) .
Ild~n~(W) (g)!I ~ c7Hgll
for any
Thus any number
w 6 An+l(X)
and
c7
22
g ~ T n + l ( x )N h 6 T
~n (x)
N
satisfy
for a n y
lld%nx(Y) (h)II >- ~71!hll
and v i c e v e r s a .
Indeed
w =
nx
y 6 Mn(X)
(y)
for some
that
for e v e r y
and
y ~ Mn(X). Applying and
(4.9)
h 6 T
to
one o b t a i n s
f = }nx
y
6 Mn(X)
N }n x
Nd~nx(Y) (h) II ~ IidCnx(0) (h) lI -
Now,
using
(3.7)
and
(3.3.3)
one o b t a i n s
~blnl lld~nx(Y) (h)II !
L~L
hFZ-F Q ( x ) ~ 2 b
b b
from
2~
IIhll.
_
Y
-
2tc 6
Thus one can take c7=2t.
2tc2 (4) b"
two t i m e s
y ~ An+l(X) , one o b t a i n s
I < Id2( nx
the f o l l o w i n g
identity
(3.3.6)
follows
from
the
inequality
" (3.3.3)
3 and
(3.3.5).
W e l e a v e the d e t a i l s
to the reader. 4.2
(4.12)
-i o ~nx(Y ) = y
nx
Now,
~bTnl
) Nhll >-
b
~ - -7
O(x)
Differentiating
(3.3.6)
where
(3.11),
that
c6
( 7
because
lld2¢nx(Y) If)llyllllhIl.
( sup h6Mn(X)
[]
We w i l l n o w s k e t c h q u i c k l y
mappings mapping
¢0,...,¢k_1 ¢ = Ck-i o...o
As for the m a p p i n g results Let
satisfies
of P a r t s M,V
satisfying
assumptions
~
implying
the C o n d i t i o n
(1.2)
be as in Sec.
an o p e n s u b s e t
all our a s s u m p t i o n s .
Nj
to
t h e n for the
of L e m m a
is s a t i s f i e d ,
3.3 holds.
t h a n all
~.
i.i. of
that if the
(1.1)-(1.3),
the e x a c t c o u n t e r p a r t
I-III can be a p p l i e d
and
let us c o n s i d e r
¢
~0
the a r g u m e n t s
For e v e r y V
j, 0 ~ j ~ k-l, ~j :N 3. ÷ V
and a m a p p i n g
Let us d e n o t e
Aj = M\Nj
for
0sj~k-l. Let us c o n s i d e r MI,...,Mk_I,
now,
M k = M 0.
k
disjoint
Let us d e n o t e
copies
of
k-i M = U M i=O l
M
noted
M 0,
and let us d e f i n e
23
on
M
the m e t r i c
~(x,y)
~
as
follows
p (x,y)
if
k-i [ d i a m ( M i) + 1
if this
Let
us d e f i n e
T
restricted
measure with
belong
to some Mi, 0 ~ i S k - i
=
_
for
In V.
is w e l l
let us d e f i n e
all
0 < ~ < I, y > 0
L Like
,y =
of
of
I
and
u(N i) = 1
submanifold
x ~ N,
V. c M. 1 1
of one
Ni"
~(Ni ) = i, 0 ~ i S k-l.
non-empty
Thus
where
in the c a s e
n ~n(~) n(Z
defined
is an o p e n
k-i N = U N. ; ~(N) i=0 i
Now, =
#
our
~t~,Y . x
( ~t
assertion
The
exact
and
counterpart
the m a p p i n g
concerning
of
T.
follows
As
24
f r o m the fact that
it is true for
We
~.
leave the d e t a i l s
to the
reader.
5.
THE F O R M U L A T I O N We
eral
formulate
form than
OF P E S I N ' S A B S T R A C T
now Theorem
in P e s i n ' s
e n c e of s t a b l e m a n i f o l d s the o r i g i n
of
~m
2.1.i
paper.
INVARIANT
from This
[Pes] 1 in s l i g h t l y
THEOREM less gen-
t h e o r e m d e a l s w i t h the e x i s t -
for a s e q u e n c e
and s a t i s f y i n g
MANIFOLD
of s m o o t h m a p s d e f i n e d
some n o n - u n i f o r m l y
near
hyperbolic
con-
ditions. Let
UI,U 2
spaces
~k
be o p e n n e i g h b o r h o o d s
and
~m-k
b o r h o o d of the o r i g i n Suppose maps
that
of the o r i g i n
, respectively. in
Then
such that e a c h
and the d e r i v a t i v e
dfn(0)
f
is a s e q u e n c e preserves
n
preserves
f can be r e p r e s e n t e d n u 6 U I, v 6 U2,
for
fn(U,V) where
=
An,B n
d~(0,0)
=0.
(AnU + a n ( U , V ) ,
Note that
some neighborhood
and
{u ( ~ k ; HuH ~ r}
by
5.1.
{fn}n~0 (i) and
the s t a n d a r d
(ii)
Let ~
~m=~ to
~m ×]Iqm-k.
(3.8).
(5.1)
= 0, bn(0,0)
= 0, dan(0,0)
are also of c l a s s Fn .
The m a p
generally
Euclidean
[Pes] I, T h e o r e m (5.1)
the o r i g i n of
Fn
C P.
is d e f i n e d
depends
= 0,
on
in
n.
B~,
2.1.1).
and in a d d i t i o n
All mappings B n U, 0 < I < min(l,u)
llAnll < l,
and
(cf.
has f o r m
bn by
of the o r i g i n w h i c h
ll'II d e n o t e s
c P ( p ~ 2)
in a f o r m s i m i l a r
an(0,0)
f0
the b a l l
THEOREM
K
an
fn o ...o
Let us also d e n o t e
where
is a n e i g h -
B n V + bn(U,V)) ,
are l i n e a r maps,
Let us d e n o t e
of
the d e c o m p o s i t i o n
In o t h e r w o r d s , Namely,
U = U1 × U2
~m.
{fn }, n = 0,1,2 ....
f : U + ~m n
in the E u c l i d e a n
are
the
invertible
norm Suppose
t h a t the s e q u e n c e
following
is true:
and there exist real numbers
s u c h that for all p o s i t i v e
integers
n,
!IBnlll-i > ~.
tn(U,V)
such t h a t
and e v e r y n o n - n e g a t i v e
=
(an(U,v),bn(U,V)).
There
exist real numbers
I < 9 < i, K > 0, and for e v e r y integer
n,
Zl,Z 2 E U
and
25
Iidtn(Zl) Then
- dtn(Z2)II
for e v e r y
numbers
C = C(0
= k~iAk,r,s,a"
also
from
U ~>2
A£ r,s,a,y e
All
sets
are
~
m U A~ . k=l K , r , s , e , y
=
and
those
For
fixed
x 6 A
let us d e f i n e
y(x)
= sup
{r ~ R;
D(~n(x),A)
Z(x)
= inf
{i > 2; -
x E A£ k , r , s , ~ , y ( x ) }"
rrs,~
~ r~ Inl
for all
integers
n}
and
As
x
6 A
£(x)
y(%(x))
In the and (6.2)
we
the
when
and
x £ A
(cf.
Theorem
.< Z(x) e - b e x p ( e
2.1
r,s
and
(3.7))
(6.l)
).
or x 6 A~ by y x 6 A£ r,s,~,y k,r,s,~,7 £(x) d e f i n e d above. and y(x)
respectively and
rrSi~
x Bm - k ( x ) r
(3.2.1),
9~(¢(x))
writino
understand
For
Bk ( x ) r and
>_ c~y(x)
future,
£
then
r,s,~,7(x)
in
r > 0
~m
by
let us d e n o t e
the
set
B(x,r)
set
eXPx(T x)-IB(x,r)
Furthermore,
for
by
x
6 A
U(x,r) . and
rts,~
n ~ 0
we
set
(Q(x) 2 b n ) 2 Un(X)
= exp
It f o l l o w s defined
by
We will x 6 A
rtsrd
V(x)
where
n
(T'n)-iB(x' ~ x ~ x from
(2.2.5)
2
that
Un(X)
c Vn(X) , w h e r e
Vn(X)
is
(3.9). construct in the
the
local
stable
manifold
V(x)
of
a point
form
= eXPx(Tx)-l(graph
Cx ) ,
(6.2)
27
and zero
%x:
k(x) ÷ Bm-k(x) B~ (x) 6 (x) '
the
radius
is s u f f i c i e n t l y s m a l l Z set of the f o r m Ak,r,s,~,y"
on any
6(X)
Naturally, equivalent
conditions
Let
and
bounded
TxV(X)
away
= Elx,
from
which
(6.4)
6.1 and
¢: N ÷ V
(1.1)-(1.3).
is
conditions
of the m a n i f o l d s
of T h e o r e m
6.1.
x (V(x)
but
= 0.
properties
formulation
that
following
= 0, d%x(0)
Subsequent
THEOREM
we e x p e c t
to the
Cx(0)
(6.3)
in S e c t i o n
be a map
Let
listed
below
in the
7.
of c l a s s
~ = ~(r,s)
0 < a < i, r . e x p ( 1 0 0 e r s ) < ~
are
V(x)
CP
(p>_2)
be a n u m b e r
such
satisfying that
10b (*)
(6.5)
I
(b c o m e s
from
(1.3)
Moreover, set
i
let
(cf.
rrs
l' (x)
2 -
The
"
(1.3), H and
inequality
from Theorem
A(x)
> _
-
1
-
,
K(x)
3.1 a n d > 1
•
v(x)
is d e f i n e d
30
< min(l,~) follows IIAnlI
immediately
and
(ii) (5.2)
from
llBnlII-I
(6.5)
follow
The i n e q u a l i t y
From
satisfies
(6.6)
(2.9).
I < ~ < 1
is an e a s y c o n s e q u e n c e
Remark.
and
immediately
follows
of T h e o r e m
it f o l l o w s
The
f rom
from
(6.5).
(cf.
(6.9)).
3.1
immediately
inequalities
for
(3.15). Inequality []
that the n u m b e r
< _ i, then
C = C(x) a new
> i. C
In-
by
C = K( 1 + ~ 2 ( I , ~ , ~ , < ) ) . Let us set 6(x)
-
r0(x) C(x)
(6.10)
Since < Q2(x) 2
r0
condition (6.4)
(6.3)
also
< 1
follows
follows
from
from
F r o m n o w on the p r o o f of T h e o r e m
2.2.1
s k e t c h y we will, case
from
(5.1.1)
and the d e f i n i t i o n
of
6(x) ;
(5.1.1). follows
[Pes] I.
exactly
a l o n g the lines of the p r o o f
S i n c e the p r o o f
for the sake of c o m p l e t e n e s s ,
in
[Pes] 1 is s o m e w h a t
give the p r o o f
in our
in full detail.
(6.1.1) defined.
Suppose
that
z 6 B(x,$(x))
a n d that
(fn o...o
f0) (z)
is
Then
(fn .... o f0 ) (z) = ~ - 4 b ( n + l ) F n x ( Z ) = -4b(n+l)
=
1
(T,n+lxOeXp-~n+l
o ~ n + l o e x P x o(Tx)-l) (z) x
31 Let now
y (V(x) .
Then by (6.2) ,
Tx(eXPx)-l(y ) = (U,~x(U)) _k(x) u 6 s6 (x)"
for some
From
(5.1.2)
we have for all
n >_ 0,
Fnx(U,~x(U )) c B(x,Q(x) e 4b(n+l)) . Moreover,
(5.1.2)
implies
that,
ll(exp:ixO ~n) (Y) II, nx 0, for
p
n > O.
J
Because
b i g enough,
x1 ÷ x
one has
and
z.
+ z, for e v e r y
i
P
and
Let
38
p(z i ,z) ~ s, P
P(x i ,x) 0
there
p(x,y)
where
the
following
reinforce-
2 I. exists
rE > 0
< m i n ( r e , R v ( X ) ) def
w = exPxl(y)
and
such
that
Re(x)
one
lld(exPx I) (Y) II
~I+~. Obviously smooth of
Starting
mapping
Let s
is a l w a y s
manifold
and
satisfied V
when
is a s m o o t h
M
is a
open
submanifold
always
the m e a s u r e
the n u m b e r s
and
e,0
Sr,s ) < 1
and
er, s)
satisfied. =
we will
that
so small
r exp(50
~,
l, k, r,
< e < i, r exp(100
such
suppose all
s
that
Sr, s)
that
for
conditions
the
from
are
fixed.
L e t us
the c o n d i t i o n s
< s
(2.5) I
< s10b
Then
(6.5) I
we d e f i n e
numbers
I,Z,K
~r,s
= r exp(3er, s) , ~ = ~r,s
Xr,s
= r exp(4er,s),
L e t us n o t e defined
preserving
us a s s u m e
rrs
Ii
this m o m e n t
satisfied.
r exp(100
are
from
~: N ÷ V
11 are
2.4. fix
(Bc)
Riemannian
M.
2.3.
Sec.
Condition
compact
that
in P r o p o s i t i o n
these 6.11
and
v
as f o l l o w s
= s e x p ( - 3 e r , s)
(2.1)
~ = Vr, s = e x p ( - e r , s)
quantities
are d i f f e r e n t
and at the end of Sec.
f r o m the o n e s
61 and d e n o t e d
by
43
the
same
symbols.
L e t us d e f i n e = ~r,s
also
= r e x p ( 7 e r , s ) , ~' - ~ = r e x p ( 8 e r , s ) , (2.2)
^
r exp(12Sr,s) J q = qr,s There
are no p a r t i c u l a r
definition
(2.1)
inequalities from
-
which
(2.5) I and X < ~ < ~' --
1
of and
X
will
be
l i m d i a m ( A r) = 0. r÷~ differentiation (see f o r
47
e x a m p l e Chapt.
l0 of
J(p)(x)
lim r÷~
[Shi] I) it f o l l o w s
that
~(P(Ar)) =
for a l m o s t 3.2.
If
all p o i n t s H
(''')H'
E
dimension,
x 6 X
and for all
is a f i n i t e d i m e n s i o n a l
then by
Let
(3.1)
~ ( A r)
Vol H
and
F
we w i l l
respectively.
Let
scalar
E1 c E
be a l i n e a r m a p p i n g .
vector
denote
{Ar}rt I.
space w i t h a s c a l a r p r o d u c t
the c o r r e s p o n d i n g
be two real v e c t o r
provided with
x - sequences
products
be a l i n e a r
volume
on
H.
s p a c e s of the same f i n i t e (''')E subspace
and of
(''')F E
and
A: E ÷ F
L e t us d e f i n e
VOlFI(A(U)) A E1
= V O l E I (U)
where
U
is an a r b i t r a r y
F1
is an a r b i t r a r y
E1
and
A(U)
Let
X
dimension on and
X
and
and
and Y
c F I.
Y
Y
3.3.
define volume by
is e s s e n t i a l l y
z 6 X
by
E1
manifolds
of the same
as
finite
Riemannian
and consequently ~y
and w h e r e
Idet A I •
diffeomorphism.
and
situation
one has
measures
metrics on
X
respectively.
a standard
=
Idet dT(z) I
one has
= Iyf d~ Y"
change
of v a r i a b l e s
s e c t i o n we w i l l
than usual.
Hilbert
d(~yOT) - - ( z ) d~ X
f ~ LI(y,~y)
'det d T ' d ~ x
For the r e s t of this
9
s u b s e t of
of the same d i m e n s i o n
3.1.
For a n y f u n c t i o n
[Rog]
F
IAIEI
elements
(3.1.2)
Let
C1
~X
For a n y
dimensional (cf.
be a
(3.1.i)
general
of
be two R i e m a n n i a n
T: X ÷ Y
IX (f°T)
This
subspace
We w i l l d e n o t e
w h i c h we d e n o t e
PROPOSITION
open and bounded
linear
Namely,
let
formula.
deal w i t h H
a slightly more
be a f i n i t e or i n f i n i t e
space.
d e n o t e the p d i m e n s i o n a l H a u s d o r f f m e a s u r e in ~P~H P for H a u s d o r f f m e a s u r e s ) . When H is f i n i t e d i m e n s i o n a l
48
restricted P with (cf
the
p
to a
p
dimensional
dimensional
[Rog]).
For
volume
U c ~P,
submanifold
(Lebesgue
by
~
(U)
of
measure)
we w i l l
~P@H
coincides
on this
submanifold
denote
its
p
dimensional
P Lebesgue
measure.
PROPOSITION f: U ÷ H
3.2.
be a
Let C1
U c ~P
be an o p e n
mapping.
If
bounded
set and
suplIdf(v) ll s d, vEU
let
then
£ ~p(U)
~ ~p(graph
f) ~
(i + d 2 ) 2 ~
(U).
(3.2)
P Proof.
If
S c U
and
if
diam(S)
r S diam{(v,f(v));v
where
the d i a m e t e r on
the g r a p h
a cover
of the g r a p h
the a s s e r t i o n 3.4.
If
then
by
E,F
where
are
=
E',
with
f.
from
every
which
to the R i e m a n n i a n cover
are
linear
the
of
/ ~
the d e f i n i t i o n
two c l o s e d
we d e n o t e
respect
Thus
by b a l l s
follows
F(E,F)
F (E,F)
of
then
E S} s / ~ r
is t a k e n
induced
= r,
metric
by b a l l s
times
in a H i l b e r t
between
E
determines
larger.
of the H a u s d o r f f
manifolds
aperture
U
and
F
Now
measure. space
• H,
i.e.
sup ( inf He-vl] ) eEE'~vEF' llell=l
F'
are
linear
subspaces
respectively.
If w e r e p l a c e
one
the n o r m
generating
of
H
the E u c l i d e a n
II'II' such
that
parallel
to
E,
F
structure
in
H
by a n o t h e r
for e v e r y
v
allvll' ~ llvll ~ bllvll' then
the n e w a p e r t u r e
a ~F(E,F)
~ F' (E,F)
Starting are
linear
PK
denotes
subspaces
of
H.
the o r t h o g o n a l
of all
satisfies
the
inequalities
< ~a F ( E , F ) . _
f r o m n o w we w i l l
L e t us c o n s i d e r manifold
F' (E,F)
the
(3.4)
consider
If
K
p - dimensional
the case
is a l i n e a r
projection following
only
of
H
metric
linear
subspace
onto 8
when of
E
and H,
K.
in the G r a s s m a n i a n
subspaces
of
H
F
then
49
8(E,F)
The
=
JJPE - PF H"
following
meaning
lemma
LEMMA
3.1.
3.5.
The main
following LEMMA
(sec Sec.
39 of
[Akh])
explains
the geometric
of t h i s m e t r i c . @(E,F)
and any
goal
of the p r e s e n t
lemma which
3.2.
such that
= max{F(E,F),F(F,E)
For
may
every
for every
also
p 6 ~
}
subsection
b e of
interest
there
exists
two H i l b e r t
Spaces
two
linear
operators
A,B:
and any two
linear
subspaces
El,
H1
H1 ÷ H2
is t h e p r o o f
the
a number
C 1 = CI(p)
and
for a n y
H 2,
with
E2 c H1
of
in i t s e l f .
JJAH ~ a,
> 0
a ~ 1
NBII ~ a
of dimension
p
one has
space
by
IAIEII JBIE2 < ClaP AB + Proof. E2
Replacing
we can assume
subspaces
and
dim H2
Hilbert
± = 0.
of t h e
Now
H2
Indeed,
choice
of
H2
complement
H1
as its
of
H1
to t h e o p e r a t o r s
Replacing
by the
by
the aperture
space
containing
HI
A ( H I)
a
2p
subspace.
in
HI"
from
if n e c e s s a r y
of
spanned
such that
l e t us d e n o t e
containing
H 1 ÷ H2
HI
d i m H 1 s 2p.
subspace
= 2p.
space
the o r t h o g o n a l A,B:
that
is i n d e p e n d e n t
us fix a l i n e a r
BJ
if n e c e s s a r y
HI A
by
UA
them.
Let
B(HI)
c H2
dimensional
L e t us d e n o t e
H2
and
of t w o
c H2'
L e t us e x t e n d to
E1
the o p e r a t o r s
by taking
and
B
I H1
by
by
A I UB
=
where
[
H1 U: H 2
÷ HI
when
H 1 = H 2 = E,
because of
A
is an i s o m e t r y ,
otherwise and
P tensor that will
E
P ~ E
E,
We
we can consider
the
can also
situation assume
A 1 = a-iA
and
to t h e
that
case
a = i,
B 1 = a-iB
instead
B.
L e t us p a s s Let
we can reduce
d i m E = 2p.
to
some
introductory
Euclidean
P A E
denote
and
power
now
be a r e a l
and
P P A E c ® E. be denoted
we
p - fold All
the
u p to the
space
its
remarks. finite
dimension
p - fold direct
exterior scalar
of
power
products
end of this
sum,
respectively. and all
subsection
m.
By
p - fold L e t us r e c a l l
the Euclidean by
(',')
norms
and
respectively. PROPOSITION
3.3.
For
any
p >_ 1
there
exists
a number
C(p)
> 0
Jl'lJ
50
such
that
for all
lleill ~ i,
vectors
IIfill ~ i,
p
1 s i s p,
one
an__~d { f i } l ~ i ~ p ' ei'
~ E,
fi
has
p
e .l -
i=l Proof.
Let
i ~ifi
~ C(P) l.= 1 ei
us c o n s i d e r
T ( h I ......... ,hp) is a
C
Value
Theorem.
onto
{ei}isis p
mapping
-f" I P P T: ~ E ÷ @ E
a mapping
=
~ h i , where h. 6 E, i=l 1 and thus P r o p o s i t i o n 3.1.
given
by the
formula
This
mapping
1 S i S p. follows
from
the M e a n p
•
Let us d e n o t e by z the c a n o n i c a l l i n e a r p r o j e c t i o n of ~ E P P A E g i v e n by the o p e r a t o r of a n t i s y m m e t r i z a t i o n . It is w e l l
known
that
i=~l ei
is e q u a l
the p a r a l l e l e p i p e d a linear
map,
spanned
E1 c E
to the
by
p-dimensional
e I , ......... ,ep.
is a l i n e a r
subspace
with
volume
Thus
if
a basis
in
E
of
C: E ÷ F
is
e I , ....... ,ep
then
i!lCei
Obviously (a) and
of L e m m a
3.2.
follows
from
two
statements
(b) :
-
B I E I I 0,
transversal
continuous.
and
x ( AI k,r,s,e,y
then for every
0 < q~,s,~,~(B)
B > 0
S1 (x) k,r,s,~,7
Iwil < - ~ ,s,~,y(B) Furthermore,
one has for
~ 1 W
there
6~ < r,s,~,72
such that for every two submanifolds
to the family
> 0, i = i, 2.
> 0
and W1
and
W2
(B)) ,s,~,y ~wi(WiNA~ ,r,s,e,y (x,q~ ,s,~,y (B)) > restricted
to
almost all p o i n t s
U(x,q~
59
y E ~i N ~ i (x Z y(B)) k,r,s,~,y 'qr,s,~, Sec.
3.1.)
satisfies
the J a c o b i a n
J(p) (y)
(See
the i n e q u a l i t y (4. 5)
IJ(p) (y) - II -< B. Remarks. It w i l l
i) m
> 0
r,s,~,y Z
follow
from the proof,
such t h a t
for e v e r y
= er,s,a, ~(B)
qr,s,a,N (B)
that t h e r e e x i s t s
B, 0 < B < i,
a constant
one can take
B
- m
r,s,e,y 2) all
If
Z
~(Ak,r, s) > 0
big e n o u g h
Let us e m p h a s i z e to the p r o o f in Sec. 4.2.
and if
a n d all
7
enough,
t h a t the s t a t e m e n t
of c r i t e r i a
of
K-property
is fixed,
then
~ ( A k , r , s , ~ , Y)
> 0.
(4.1.3) given
Let us d e s c r i b e
the g e n e r a l
in the p r o o f of s t a t e m e n t from e s t i m a t e s
obtained
of s t a t e m e n t
Indeed,
roughly
the m a n i f o l d s
is fully s u f f i c i e n t
structure
of the proof.
that all the d i f f i c u l t i e s
(4.1.2).
Statement
(4.1.3)
in the p r o o f of s t a t e m e n t
(4.1.1)
speaking, ~n(~i),
i n s t e a d of w o r k i n g the time
(Secs.
also
follows
for some
i = 1,2,
n
in
directly
easily
(4.1.2).
the m a p p i n g
(4.1.2).
parts w h i c h
to the
family
of T L S M in P a r t
~,
are d e f i n e d
follows The
one can d i v i d e
into c o r r e s p o n d i n g
which
o (~)-i
we w o r k a l m o s t ~'0w = T~(w ),
I, all
o
in some n e i g h b o r h o o d s
~m.
A characteristic frequent verifications
f e a t u r e of the p r o o f that v a r i o u s
a r e in fact w e l l d e f i n e d . paper
with
are c o n c e n t r a t e d
from the p r o o f of big enough,
5 - 9) w i t h the local m a p p i n g s
o e x p ~-i (w ) 6 ~ o eXPw 0
[Pes] 2 and r e p o r t e d
in
s a t i s f y the a s s u m p t i o n s of (4.1.2) w i t h r e s p e c t Cn l (x)) . (Rk,r,s,e,y M o r e o v e r , as in the p r o o f of the e x i s t e n c e
of
for
13.2.
F i r s t of all let us n o t e
proof
e, 0 < e < 1
small
as w e l l
as in P a r t
is the n e c e s s i t y
composition
This problem I, but h e r e
already
of m a p p i n g s , occurs
of etc.,
in the P e s i n
it is one of the l a n d m a r k s
of
the f i r s t half of the proof. To p r o v e T h e o r e m
4.1, we w a n t
to c o m p a r e
~ I(D) W
where
D
is a m e a s u r a b l e
s u b s e t of
of the p r o o f w h i c h g o e s b a c k to Ya.G.
~i
with
~ 2(P(D)), W
N AZ . The m a i n k,r,s,~,y S i n a i (see [Ano] 2) is to
idea
60
deduce the comparison comparison between
between
~
Cn(wl )
VwI(D)
(¢n(D))
and
and
v
~w2(P(D)) ~n(w2 )
from the
(¢n(p(D)))
for
n
big enough. Indeed,
as the submanifolds
family of TLSM
SZ (x), k,r,s,~,y
~n(~l N ~£k,r,s,~,y (x,q)) and closer as
and
n ÷ ~.
between
and
W2
~n(~2 N A~k,r,s,e,y(x,q))
in the integral,
~wI(D)
are transversal
to the
then one hopes that
And really it is so.
of the change of variables comparison
W1
and
become closer
After using the formula one deduces the desired
~w2(P(D)).
Unfortunately,
the
realization
of this idea is not easy. First of all for n big ^Z enough the set ~ n ( ~ i N Ak,r,s,~,y(x,q)) in general does not belong to one local chart.
Thus,
~n(~l N AZ (x,q)) k,r,s,~,y such that for every chart.
Moreover,
to avoid this difficulty,
by sufficiently
i, 1 5 i ~ p(n),
we cover
small pieces
W~n
belongs
{W 1 }-< < in ±_i_p(n)' to one local
the next step consisting in comparison between W1 and Cn(w2) (#n (p(D N in))) is also rather
~ n(wl ) (~n(DN W~n))
complicated. The realization
of the above idea is contained
and is divided between eight Lemmas The proof is sharply divided formed by Secs.
5 - 7, contains
to the proof properly After contains
speaking,
5 - 10
in two parts.
The first part
all necessary tools and preliminaries which is given in Secs.
some important preliminaries
described
among others Lemma I, we prove in Sec.
which can be considered
in Secs.
I - VIII as follows.
as the Main Lemma.
8 - 10.
in Sec.
5, which
6 the Lemma II,
Roughly
speaking,
this
Lemma describes the asymptotic behavior of #n(A), for large where A is a small piece of ~i whose size depends on n. Sec.
7 is mainly devoted
construction mentioned
to the careful description
which leads to the small submanifolds
of the
{W~ } in l~isp(n)
above.
In order to be able to estimate special coverings these coverings In Sec.
n,
of
#n(D).
9 n(w2) (%n(p(D))) ,
We devote Sec.
8 to the construction
and to the proof of their good geometrical
8 we prove Lemmas
III-
V.
as a second main lemma and Lemmas
we need some of
properties.
Lemma III can be considered IV and V are devoted to the con-
61
struction of desired covering.
Unlike all other sections,
Sec.
8
has a true g e o m e t r i c a l character. Sac.
9 is m a i n l y d e v o t e d to the proof of third m a i n lemma
Lemma VI giving the e s t i m a t i o n w h i c h allows to c o m p a r e ~w2(P(A)) 9 n(wl)(¢
n
for
A c ~i
(A)) and
as above,
~ n(w2)(¢
n
~wI(A)
and
if only one knows the m e a s u r e s
(p(A))).
A m o r e global Lemma VII is an
easy c o r o l l a r y of Lemma VI. Finally,
in Sac.
i0 we prove T h e o r e m 4.1.
special coverings whose c o n s t r u c t i o n in Sac. 8.
The proof uses some
is based on coverings c o n s t r u c t e d
The proof uses Lemma VIII w h i c h is the a d a p t a t i o n of
Lemma VII to the elements of just c o n s t r u c t e d coverings. note that the exact values of c o n s t a n t s at the end of Sac. Sac.
i0.
ii contains a very i m p o r t a n t c o n s e q u e n c e of A b s o l u t e
C o n t i n u i t y T h e o r e m 4.1. (see Sac.
Let us
from T h e o r e m 4.1 appear only
ii.i),
N a m e l y we show that the conditional m e a s u r e s
induced on T L S M
~V(Y) N U ( x ' q ~ , r , s , ~ , y ~
~
Z
Y6U(X'qk,r,s,a,y)
by an
N A£
k,r,s,e,7
a r b i t r a r y measure w h i c h is a b s o l u t e l y c o n t i n u o u s with respect to the Riemannian measure measures
~,
are a b s o l u t e l y c o n t i n u o u s w i t h respect to the
induced on these submanifolds by the R i e m a n n i a n metric
The idea of the proof goes back to Ja.G. The role of L y a p u n o v m e t r i c c o m p l e t e l y illusory. the sets
U(x,q)
Sinai
(see
p.
[Ano]2).
in the f o r m u l a t i o n of T h e o r e m 4.1 is
In fact this m e t r i c intervenes o n l y through
and by the q u a n t i t i e s
IWII
and
IW21.
But as on
the set
A£ the L y a p u n o v and the R i e m a n n i a n m e t r i c s are r,s,~,y' u n i f o r m l y equivalent, nothing will change if instead of L y a p u n o v
metric we use to formulate the T h e o r e m 4.1. the R i e m a n n i a n metric. As far as the proof of T h e o r e m 4.1 is concerned, metric plays a substantial role.
the L y a p u n o v
In p a r t i c u l a r this is true for the
Main Lemma II, but also for m a n y other places.
As it was e x p l a i n e d
in Part I, the use of L y a p u n o v m e t r i c allows to reduce the study of mapping
~IA i
to the study of m a p p i n g
satisfying the u n i f o r m
rts hyperbolic conditions Finally,
(see
(2.2.3)i).
let us u n d e r l i n e that A b s o l u t e C o n t i n u i t y T h e o r e m is
not a local result despite the fact that it is c o n c e r n e d w i t h a small n e i g h b o r h o o d of a point S£ (x) k,r,s,~,y
x.
Indeed,
involves the m a p p i n g
the d e f i n i t i o n of the family ¢
on the w h o l e
N.
62
5.
START OF THE PROOF - I
5.1.
Starting from this moment up to the end of the proof of
Theorem 4.1,
x ( AZ k,r,s,e,y
will be fixed once and for all. w 6 Ak,r,s,~, Y n U
Let us consider a point US note
V(w) = (TwI
Let
US
o
exPwl) (V(w)) .
r,s,~,y '
2
~£ (Tw ° exPwl)(Y)(9(w)QB (~r's'~'Y) (see2
As follows from (6.1.i)
I
Let "
£
I y E V(w) N U lw, 6r's'~'Y2 ) "
choose a point
P0 = (u0'v0)=
x
Let
(2.10)). (5.1)
and from (2.9) for every
n > 1
one
has I P n = (Un,Vn) def ( ~
~i) (P0)
= (T~n(w)
o
i)
exp n
(}n(y) )(
(w)
(5.2))
( B n (w) , Bn,w.[)
where
is defined in Sec. 2.6.
Moreover, the proof of (6.1.i) I together with (2.9) gives [ (Un'Vn)N = [I( n-~ 9! .= 1 I~
) (p0) II ~
200(K r t s)nll (u0,v0)ll
n02(Z,s,r,a,y)
one has
]n < _i [Q(Z,a,y)a2bn]2.
[a8bexp(_15Er,s)
/Y Let us note that in fact this proof (under assumptions of Lemma
5.1.2)
as shown
in Fig.
of Lemma
one has
5.1.1)
~2(Vn,6,)n
and for
5.2.
Lemma
permits
and
us
to
~ 2 ( V n , 6n )
B 2 (qn) n
with respect
B2(qn ) n
consider
= Bl(qn ) n
consider
(under assumptions
to the standard
is, generally
speaking,
Euclidean
an ellipsoid.
5.1.i will play in the future only a technical
The role
and
~P,
v ~ B2(Vn,~ ~)
Bn(qn)
to
/
Considered
norm in
n > 0
n t nO
2.
~
2.
that for
c Int Bn(qn)2 = {v 6 ~P; llvll < qn }
/
Fig.
implies
for
any
because
n ~ n O,
for
~'(u,V)n
n ~ no,
~'n
for
role.
It
u 6 Bl(qn)
is well defined
on
x B~(qn ) c Bn(W ) . of
far
(5.1.2) every
v 6 B2(Vu,~ ~)
will n ~ 0
because
be the
completely mapping
for every
analogous. ~(u,v)
n ~ 0,
It
for !
~n
permits
u ( {i (qn)
is well defined
85
on
Bn(qn)
= B ~ ( q n ) × B 2 ( q n ) c B (w). n n
f r o m the fact that for e v e r y =/~q0[ from
6.
8bexp(_15Sr,s)]n (6.7) I b e c a u s e
n
t 0
The last i n c l u s i o n has
< [Q(/,~,y)
/2 qn =
2bn]2.
q0 ....r,s,e,y2
0
=q(/,r,s,~,y,C)
q(l,c)
(cf.
(2.1)
Sec.
i.i I,
and
b
be some number.
Let us d e f i n e
r,s,~,y ~-~ 8q(Arl,s+l ) ' F
~n-I ' F(CI_+n)
q(l,C)
=
by
= min
- (2.3)) , w h e r e
q
comes
(6.1)
from the c o n d i t i o n
(Bb)
from
HA Z br,s ' H c o m e s f r o m the T h e o r e m 3.1 I y f r o m c o n d i t i o n (1.3)I. F r o m (2.4) , ~ > ~ and f r o m
F = F(/,r,s)
= 2
^
comes
(2.5),
~
> i.
Thus
q(l,c)
the f i r s t s t e p t o w a r d s Theorem 6.2.
The d e f i n i t i o n of n u m b e r s
of q(l,C) 1 qr,s,e,y(B)
is from
4.1.2.
The
following
first p a r t d e s c r i b e s where
> 0.
the d e f i n i t i o n
W
First Main Lemma the b e h a v i o u r
is a s u b m a n i f o l d
in the L y a p u n o v m e t r i c case w h e n
W
to d e s c r i b e
is v e r y c l o s e the behavior
can be c o n s i d e r e d l-lemma
close
is a b o u t
in o u r
(see for e x a m p l e
p a r t is u s e d
consists
of Cn(w)
to the s u b s p a c e 6'. n
~n(w)
framework [New]
in the p r o o f of
or
n
The
big e n o u g h ,
E2W,
whose
size
The s e c o n d p a r t d e a l s w i t h the
to the s u b s p a c e
of
of two parts. for
for all
E2W.
T h e n we are able
n ~ i.
as a c o u n t e r p a r t [Pal]).
(4.1.1),
This
of so c a l l e d
Like Lemma
the s e c o n d
lemma
5.1,
the f i r s t
in the p r o o f of
(4.1.2). LEMMA
6.1
(6.1.1).
(II) Let
to the f a m i l y
x ~ A k1 , r , s , e , ¥ S kZ, r , s , ~ , y(x)
L e t us c o n s i d e r such that T L S M
V(w)
a point
and let _in _
W
be a s u b m a n i f o l d
U(x,~ / r,s,~, y).
l
w E ilk,r,s,~,y n Int U(x,
intersect
W
at the p o i n t
transversal
z 0.
6r's' ) ~ 2' Y Let us n o t e by
88
(u 0 I v 0) Thus
6 ~m
for
= ~k
£0 > 0
® ~P, small
the point
enough,
there
(u0,v 0) = T'W o e x P w l ( z 0 ) " exists
the u n i q u e
C1
~0: B2(v0'A0) + ~ k such that ~0(v0) = u 0 and that , -i ~2 eXPw o (~w) {(Y0(v),v) ; v ~ (v0,A0) } is a s u b m a n i f o l d Then (where > 0
there
no
comes
such that
of class
exists
C I,
a number
from Lemma
for e v e r y
n I = nl(l,k,r,s,e,y,W)
5.1),
n t nI
~n: ~ 2 ( V n , ~
a number there
) ÷ mk ,
~nl
exist
=
of
W.
t nO
(3.13) I
0
6 IRk , v 6 ]Rp. (2.2.3) I
(3.14) I
and from the
one has for
u 6 ]Rk (6.12)
llBn(V) II >- ~llvll
PROPOSITION
o ~ o eXPn
(AnU+an(U,v),BnV
Let us d e n o t e definition
that
-i eXPn+l
o
lld~n(Vn) II.
6.1.
n ~ nll
There one has
for
v 6 ]Rp.
exists
nll=
n l l ( l , r , s , e , ¥)
such t h a t f o r
69
lldtn(Un,Vn) II < exp(-10nEr,s). Proof.
As
n ~ 1
(u0,v 0) 6 v(w),
then from
one has
(Un,V n) 6 Bn(W).
3.1 I.
From this theorem,
Theorem obtains
(5.2) one knows that for every
Thus for every
n ~ 1
from
dtn(0,0)
(5.3), as
we can apply = 0
one
IIdt n (Un,V n) II = IIdt n (Un,V n) - d t n (0,0) II -
0
IiKn+tiI-< IIKnlI~t. Thus,
to finish
the proof
Q(l,k,r,s,e,y,W)
it is e n o u g h
= supNd~nl2(Vnl2)II
< +~,
for all
71
where
sup
is t a k e n o v e r all p a r t s
w E iI k,r,s,e,y
N Int U ( x
61 r,s,~,y ' 2 )"
the fact t h a t for a n y f i x e d and the d e c o m p o s i t i o n Indeed,
one can a p p l y
Beginning n
> 0
max
q > O,
But this
the a b o v e
remarks
for all
where
is so in v i r t u e
~q(AS,r,s,a,y)K
TwN = E l w ~ E 2 w
f r o m now,
so small
Z 0 ( W N V(w),
is a c l o s e d
is c o n t i n u o u s to
n > n12
of
on
set
that set.
g = n12. we fix the n u m b e r s
that
]Idtn(~n(V)
I v)]l
< exp(-10ne
r,
(6.18)
s ),
v6!82(Vn,~ n) n-nl2 max
I[d~n(V)[I ~ 2Q~
(6.19)
[I~
(6.20)
vEiB2(Vn,~n) and t h a t max
n
(v) II < in.
vEB2(Vn,6n ) This
is p o s s i b l e
the s e c o n d of
by P r o p o s i t i o n
n t nI
6.2 and
(5.3)
together
n I = nl(/,r,s,e,y,W)
t n12
s u c h t h a t for
one h a s
1200 H A / r,s yb~
(~,)n < e x p ( - 1 0 n s
r,s
)
(6.21)
< ~ - e x p ( - 1 0 n e r,s)
301(~,)n
and
2Q~
nI
This n ~ nI
with
(6.13).
L e t us fix the n u m b e r all
6.1 and
nI
n ~ n I, (6.2)
(6.2)n , (6.4) n and
one knows
n
that
v E B2(Vn,~).
v I ,v 2 E B 2(Vn,~n ' ) .
and let
Then
-lOne lltn(~n(vl),vl) Proof.
- tn(~n(V2),v2)l]
By the Mean Value
Theorem
< e
r'Slivl-v21I.
one o b t a i n s
that
IItn(Yn(vl) ,v I) - t n ( Y n ( V 2) ,v 2) II _
3d n-
that
for e v e r y
and e v e r y
n t 0
> 3d n
one has (8.2)
c Vn(W )
K(z,q)
(3.9) I • Let
1.1 I, 0 < R < 1.
n ~ 0,
(8.1.1)
Proof.
from Sec.
= {y 6 M;
y 6 K(z,3dn).
p(y,¢n(w))
O(y,z)
~ q}
Thus
~ 3d n + p(z,¢n(w)).
and w h e r e
Vn(W)
is d e f i n e d
by
90 By
(2.9) and
(6.10)
0(z,#n(w))
n
one obtains that
_< /2 [IzlI _< /2 (ll~n(Vz)]! + NVz]I) _
0 disjoint
for some
and of P r o p o s i t i o n
d Q(z, ~ )
exp -in
d i e m Qj ~ 2dn,
n o t e d by K' (n,j),
Let us c o n s i d e r
of
1 i i ~ n 5.
Qf n Q(zj,3dn) } def YJ ~ K'
n u m b e r of m u t u a l l y
Q(zi,3dn),
As
Q(zj,dn),
all the i n d i c e s
and a n u m b e r
1 s j ~ n7,
#{i s f ~ n7;
that there
> 0
L = K + i.
i = i(j),
< as < _ n7
1 s k ~ s.
with
(8.33)
let us c o n s i d e r
for some
1 s a I < a 2 < .....
by
(8.4.3)
1 S j ! n7,
_< K-
z,
(8.34)
it is s u f f i c i e n t
that for open balls
n
and
d Q(z,~)
and
Q(z,
j
K".
c Q(zj,3dn).
d ~)
Q(z
to p r o v e
as above, the d n --3- c o n t a i n e d
of r a d i u s
does not e x c e e d
8.1 one o b t a i n s
is w e l l d e f i n e d
(8.34)
,
In v i r t u e
d -~ ) c Wn(wi,Y i
c Q(zj,3dn)
c
[~n(Wi,Yi,6n)] " (w i ) As
Q(z,
d ~)
from Proposition
c Q(zj,3dn) 8.1.1.
and as
one o b t a i n s
zj
that
( ~n(Wi,Yi,{ z ( Wn(wi,Yi,
6~),
then
3 , ~ ~n ) .
Now,
n
107
d from Lemma
8.1 applied to
respectively,
d ~2(~,n T )
n > n6
is defined
immediately
_
0
(T k
o exp -I ) (~k(zl)) ~ (w)
(w)
=
~'. ]
'=
(p)
(9.4)
--i --i (Uk,V k) .
= Let us notice
that these
points
actually
depend
on
n.
w(k) ~ Pk
(u ,v k
p2
(u2,v 2)
Pk=(Uk,Vk) M
Fig.
8.
F w(k)
=
w(k)
=
2 Mk -i ) (¢k(w)), (T' o exp k #k (w) ~ (w) -i ) (~k (Q)) (T' o exp k Ck (w) ¢ (w)
i = Mk (T' o exp -I ) (Wk) , i = 1,2, ~k (w) ~ (w)
LEMMA
9.1.
For every
n8(i,r,s,e,y,t)
> 0
C > 0 such that
zl ( #-n(~ )i n A(x,C) (cf.
(2.2)
and
(2.11)).
__if
and
t > 0
for every
II~In - Vnllnl _< ~ n
there
0 _ n8(t) then
n8(t)
=
and --2Vn 6 B 2n(v2, (C+t)~n)
111
Proof.
Using
(5.4)
as well
as
(2.11)
we have
f
r~n Vn~l + HVn~ vn~r + rv~ V~nr< V~n Vnr~< ~ ~n
2 2 n + ]](u0,v 0) II) ( 0
and
one has
IVOI ((~mj) ~n)
10.2.
0 < ~
1, -
that
there
the covering of
exists
some n u m b e r
~1 by t h e s e t s n
L > 0
120 I
{D~j}l!j~n 7
is of m u l t i p l i c i t y
at m o s t
L.
We will
denote
this
I~mSN. 3 covering 10.2.
by
A.
We will
describe
has m u l t i p l i c i t y
one,
n o w h o w one can c h o o s e
except
it we p r o c e e d
consecutively
Q(Zj+l,2dn) ,
j = 1,2, ..... ,n 7 - 1
all sets
D mj+l 1
the c o v e r i n g
of
~i n
the m u l t i p l i c i t y
N DI
U i=l i
=
~l
c
n
For
The
10.1
every
e,
N
D m1 j + l
Q ( z j , 2 d n)
and in
(j+l)th
j Nk c k=l U m=l U Dmk"
f o r m e d by all r e m a i n i n g
of c o v e r i n g
N
{ D Ni } l1! i S
wh$ch
To o b t a i n
to the b a l l step we e l i m i n a t e 1 {Di}12i2 N
Let
elements
of
is not b i g g e r
N.
be
Clearly,
than
L
and
(10.6)
NU D I c ~1 . i=l 1 n
(10.7)
we w i l l n o t e
following
LEMMA
of
n 7 Nj D 1 U U mj ' j=l m = l
D~l = D mj
the p r e v i o u s
f r o m the b a l l
such that
a subcover
a set of a v e r y small m e a s u r e .
Lemma
lemmas
(VIII).
6 i = nmj' ^
~2I = ~2mj
etc.
is the last of our e i g h t b a s i c
it is e s s e n t i a l l y There
0 < 8 < m i n ( ~1,
exists iC2)
(see
(10.2)).
lemmas.
Unlike
global.
C 5 = C 5 ( Z , k , r , s , ~ , Y)
(cf. L e m m a
8.2),
such that
and for e v e r y
for
n ~ n4(%)
1
i~iVO (¢-n (DI)) N
- 1
comes
f r o m Sec.
(10.8)
~ C5(0+C),
I #-n v0( ( U D )) i=l
where
C
Proof.
Let us d i v i d e
G
B
and
(G=good,
7.4.
the set of
B=bad)
{i .... ,N}
into two d i s j o i n t
subsets
121
l i 6 G
iff
i 6 B
D i1 E Q(zj,2(l-0)d n)
j
otherwise.
It is clear that if Moreover, (where
for some
n4(@)
i E G,
]
I
then
Int(D~ N D~) = ~ for every i # j. I 3 it follows from Proposition 8.2 that for n ~ n4(@) ~ n 3
arise from Lemma 8.2) and for
1 5 i ~ N
one has
I
diam D~ ~ 28d . 1 n such n
Thus,
from Proposition
i0.i one deduces that for
n7 U D~ c U {z E Q(zj,2dn) ;~(z,~Q(zj,2dn)) i6B 1 j=l where we note by metric
p
on
~
the metric induced on
~n(~l)
(10.9)
by the Riemannian
N.
We will denote Aj(@)
~ 4@dn},
A(zj,28)
defined by
(8.14) by
= {z 6 Q(zj,2dn) ; ~(z,~Q(zj,2d n) ~ 4@dn},
Since the multiplicity
of covering
{Di}l~i~ N
Aj(8),
1 ~ j s n 7.
does not exceed
one has
f i~190Ni ( _ n ( D ! ) )
=
~ i6G
i(_n(Dl))
+
[ vl(_n(Dl)) i6B
_
_ m a x ( n 2 ( 8 ) , n 3 , n 4 ( 8 ) , n 9 ( 8 ) )
one d e d u c e s
,
that
one has
-n
v0(¢
(Aj(8)). -< C 4(c28
1
+ C).
v0 (~-n (Q (zj ,d n) ) This
proves
10.3.
Now,
i0.i)
that
(10.12)
and c o n s e q u e n t l y
it follows for all
p(D 1 n A(x,C))
from L e m m a
i,
(10.8).
9.1 a p p l i e d
1 ~ i } N,
(p
to
denote
t = ~
(cf. Sec.
the P o i n c a r 4
c ~2
map) (10.13)
1
if only Lemmas
n ~ nl0 = n10(~,8) 8.3,
D = D(P,h)
~n(p(D)
8.4 and c ~i
(10.7)
one o b t a i n s
(cf. Sec.
= p(#n(D))
de__~fm a x ( n 4 ( 8 ) , n 8 ( w ) )
8.5)
c p
that
for
from now,
C,
i=UlD
8,
Consequently,
n > nl0
by
and for
one has
=
U
p(D
)
i=l Beginnlng
"
610
and
c
U
i"
(10.14)
i=l 620
will
be a s s u m e d
so
124
small
that all our p r e c e d i n g
those
enumerated
we will
suppose
assumptions
in c o n s e c u t i v e
lemmas,
about
them,
in p a r t i c u l a r
are satisfied.
In a d d i t i o n
that
8d 0 0 < 03 < - ,
PROPOSITION nll(e)
10.2.
such that
2 --2 ~n (Di) 1 1 ~n(Di )
Proof.
(10.15)
There
exists
for any
a constant
e,03,n ~ nll
03
1
Using
and
and
Vn(Di)2 --2
i, 1 ~ i ~ N,
(I0.16)
introduced
Vol((Di)03
in
(10.2)
one can w r i t e
V°l((Di)03n)
)
V o l ( D i)
n V o I ( D i)
(10.17) MI(DI(e))
1 1 Mn (Di (e))
product. exists
now e s t i m a t e
From
condition
nll I = n l l l ( @ )
1 - e -
0.
This a s s u m p t i o n becomes
I n c i d e n t a l l y the i n v a r i a n c e of
never d i r e c t l y appear in the proof of T h e o r e m 4.1. 10.6.
We pass now to the proof of
(4.1.3)
i.e. to the e s t i m a t i o n
•
128
of
IJ(p) (y) - 1 I . To this end we w i l l
to all our p r e c e d i n g
0 < C < ~
Let
i
transversal
As Fubini
C
so small
that
in a d d i t i o n
.
(10.28)
x 6 A£ k,r,s,a,7
For a n y
W6 =
consider only
conditions
be a d e n s i t y
6, 0 < 6 < q(2,C) to the f a m i l y
p o i n t of the set
let us d e f i n e
SZk,r,s,e,y(x) ,
A£ k,r,s,a,¥"
the s u b m a n i f o l d
W~,
by the f o r m u l a
! (eXPx o (Tx) -l){(6,v) ; v 6 B 2 ( q ( 2 , 0 ) ) }.
x
is a d e n s i t y
Theorem
p-dimensional.
p o i n t of the set
we can find Riemannian
and consequently
A kZ, r , s , ~ , y ' then by the Z 0 < ~ < e such that the rwsi~,~
6,
volume
V w ~ ( W 6 N A(x,C))
V w d ( W 6 N A(x,C))
Let us c o n s i d e r
the m a p p i n g s
> 0
(cf.
(8.24)). and
P~I,~
is p o s i t i v e
p~
~2"
Clearly
W~,W
P = P~I,~2 = ~ 6 ' ~ 2 As
VW6
° P.~iw,w 6~
(W~ N A(x,C))
> 0
and as m a p p i n g s
p~ ~ i = W~,W
and
P~i ~ are a b s o l u t e l y c o n t i n u o u s in v i r t u e W ,W 6 one o b t a i n s t h a t v. (~i N A(x,C)) > 0, i = 1,2.
of
( ' P~Z,~
)-i
(4.1.2) , then
1
Let us n o t e b y that
w
T
the set of all p o i n t s
is a p o i n t of d e n s i t y
to the m e a s u r e the set
~i
~2 n ~(x,C)
~l-almost
all p o i n t s
with of
absolutely
continuous,
belong
p(T).
to
L e t us c o n s i d e r the p o i n t s of d e n s i t y s u c h t h a t for e v e r y
of the set
a n d that the p o i n t respect
y 6 T.
are
v2- almost It f o l l o w s
t h a t for e v e r y h,
~ i N A(x,C)
p(w)
0 < h < h(6),
of
v2.
density
all p o i n t s
of
p o i n t of
As and as
there exists
o ne has
p-i
is
~2 N A(x,C)
from the d e f i n i t i o n
e > 0
such
with respect
is a d e n s i t y
to the m e a s u r e
~ i N A(x,C) then
w E ~ i N A(x,C)
of
h(e)
> 0
129
~l(Q(y,h))
Now,
_< (I + e ) ~ I ( T
(10.27)
and
v2(P(T N Q(y,h)) i.e.
(10.29)
NQ(y,h)).
(10.29)
imply
that
if
0 < h < h(e),
one has
_ 2C8C 0 .
Sr,s,~,y(B)
(c o )
This
finishes
the p r o o f
ii.
ABSOLUTE
CONTINUITY
ii.i.
The
(Theorem of
aim of
ii.i)
which
is due
a n d of F u b i n i
asserts
that
11.2.
Let
the
in fact
on
(X,~,o)
sigma-field
Roughly
are
subsets
of
Sinai
speaking,
space, X
an
important
(cf.
this
where
on w h i c h
5
Continuity
theorem
on local
continuous
theorem
Lecture
of the A b s o l u t e
induced
absolutely
be a L e b e s g u e of
is to p r o v e
to Ja.G.
measures
N,
MEASURES
consequence
Theorem.
the c o n d i t i o n a l measure
section
is an e a s y
Theorem
4.1.
OF C O N D I T I O N A L
the p r e s e n t
[Ano] 2) and w h i c h
by a s m o o t h
of T h e o r e m
stable
manifolds
on them.
we n o t e
as u s u a l l y
the p r o b a b i l i t y
by
measure
is d e f i n e d . A partition
B
X/B
is a L e b e s g u e
into
a family
of
X
is c a l l e d
space.
Then
of c o n d i t i o n a l
measurable
the m e a s u r e
measures
Oc'
if the
~
factor-space
can be d e c o m p o s e d
c E 6,
so that
= F o . Jx/B e These detailed
measures
discussion
conditional For
E X
definition PROPOSITION measurable
ii.i.
absolutely
CB(x)
Let
~
almost
measurable
zero.
partition
the e l e m e n t
of
B
is a s t r a i g h t f o r w a r d
(X,~,~)
of
~- m e a s u r e
For
and
containing
corollary
x.
of the
measures.
X.
all
with
be a L e b e s g u e
Let
continuous
continuous
spaces,
denote
proposition
partition
for
up to a set of
[ROC]l_ 3.
of c o n d i t i o n a l
absolutely
Then
see
let
following
unique
on L e b e s g u e
measures
x
The
oIn_n M,
are
~
with x
respect
E X,
respect
space
be a n o t h e r
to
t__oo ~
the m e a s u r e aCB(x )
and
let
probabilitz
and
so t h a t gCB(x )
B
be a
measure d ~ do _ f. is
131
d~c 8(x)
f Cs(x)
~C~(x)
C
(x) fd~c B(x)
8 11.3.
Beginning
assumptions measure
of
f r o m n o w we w i l l
(4.1.3)
i n d u c e d on
X c N,
normalized
q rZ, s , ~ , y ( B )
If compact
0 < ~(X)
measure
6
~ ~(X)
submanifolds, measure
its m e a s u r a b l e
Let us n o t e by
z (B(q(B)), on
@
and
then by
vx
~ P.
denotes
the
For the sake of
i Sr,s,~, Y (B) = e(B). we w i l l d e n o t e
the
X.
~,
of
X
x ( X,
f o r m e d by the smooth, we w i l l d e n o t e
i n d u c e d by the R i e m a n n i a n B(q(B))
: BI(q(B))
the
metric
x ~2(q(B))
P.
c m m
and
~ = {y x ~ 2 ( q ( S ) ) } y 6 ~ l ( ~ ( B ) ) . the n o r m a l i z e d
Lebesgue
measure
measure
with the normalized
let us c o n s i d e r
one d e d u c e s
I
= q(B)
t h a t the c o n d i t i o n a l
coincides
def ~.
the m e a s u r a b l e
From Proposition
measure
a l m o s t all
¢
D
metric
on
B(q(B)).
pC~(z ) ,
p-dimensional
Lebesgue
C~ (z). n
U(x,q(B))
v
C6(x)
implies
the c o n d i t i o n a l for
As b e f o r e ,
partition
the set
partition
Fubini Theorem
of
on
then by
on
Let us c o n s i d e r
Now,
< +~,
is a m e a s u r a b l e
normalized
measure
fulfilled.
by the R i e m a n n i a n
b r e v i t y we w i l l n o t e If
are
N
s u p p o s e o n c e and for all t h a t the
v~ (y)c
y ( U.
depends
o n l y on
ii.i
q =
it f o l l o w s
is e q u i v a l e n t
Moreover,
that there exists
partition
directly
to the m e a s u r e
from condition
a positive
l (exp x o (Tx)
function
%
(Bc)
-1)~
that ~N Y
f r o m Sec i.i I
such that
i, r, s, ~, y,
(ii.i) is d e f i n e d
for
0 < t _< i,
lim $(t) t÷0
= 0
and that
d~)~n (y) 1
< @ (B) .
(11.2)
132 11.3.
Let us The
recall
following
that
A(x,C)
Proposition
is d e f i n e d
is a d i r e c t
by
(8.24).
consequence
of C o r o l l a r y
7.11 •
PROPOSITION
11.2.
If
y ( ~. ( X , T
(T'x o exp~l) (V(y)
n U(x,q(B))
where
~y: BI(q(B))
11.4.
Let
A c C
÷ B2(q(B))
(x) n A (x , ~ )
be a m e a s u r a b l e equivalently •
subset
then
= {(U,¢y(U)) ; u ( B I ( q ( B ) ) is a
C1
},
mapping.
)
of
(11.3)
C
of p o s i t i v e
Let us note
(B)),
(x)
of p o s i t i v e
(x) - m e a s u r e
~C
or
v ~ - measure. x
X = {z (A(x, q(S)) ; V(z)
n A ¢ @}
and ^
[A] =
U (V(z)
n U(x,q(B)).
z~X Let us c o n s i d e r {V(z)
the p a r t i t i o n
R U ( x , q ( B ) ) } z ( ~.
Theorem ~([A])
and > 0
this
and that
~
from now,
last a s s u m p t i o n
Under
ii.i.
For
is e q u i v a l e n t function C~ (a)
~
Continuity
Theorem,
is a m e a s u r a b l e
partition that
suppose
Fubini
[A] is m e a s u r a b l e ,
we will
~ -almost
to the m e a s u r e satisfying
1
[A] into the sets
in
of
[A].
(4.1.3),
any r e s t r i c t i o n
that
q(B)
= e(B) ;
of g e n e r a l i t y .
we will n o w p r o v e every v ~. a
a ( [A], Moreover,
(ii. i) such that
one has
dv [A] C___i(a) I d~a
of
sees that
does not p r e s e n t
this a s s u m p t i o n
THEOREM
From Absolute
(7.1.3) I one e a s i l y
Beginning
~
_< ~ (B).
the m e a s u r e there
exists
~ a -almost
~)[A] C~(~) a positive
everywhere
on
133
Proof.
Step
1
For
"
a 6 A£ k,r,s,~,y
and
n U(x,q(B)
K c V(a),
we d e n o t e
K(D)
=
U C z6K
(z)
and
~a(K)
= ~[A] (K(n)).
(11.4)
Y
K(T]) /
~
Fig.
Let ( A c C will
use
write
s ( [A]. n
(x) ; y
In v i r t u e every
y
and
(y,s)
and s
s = Cq(y) s
are
of
the P o i n c a r 4
n
n C~(s),
uniquely of
where
determined s 6 [A]
y 6 V(x) by
s.
and
Thus
and we w i l l
we
sometimes
s.
of P r o p o s i t i o n
A = Cq(x)
9
as c o o r d i n a t e s
instead
y 6 C~(x)
Pxy:
Then
I
" Cn ~ (Y) )
11.2 map
[A] -~ Cr](y)
N
and Pxy
[A]
(11.3) def =
PC
it f o l l o w s (x)C
(y)
that
for
134 is w e l l d e f i n e d . L e t us d e n o t e
sets
{Cq(y)
Let
Q
by
~A
the m e a s u r a b l e
[A]}yE[A].
y 6 C~(x)
and
L e t us d e n o t e
s (Cq(y)
N
partition
also
[A].
of
[A] into the
vqA = i ~q. y vq (C n (y) N [A] ) Y Y
We d e f i n e
d~ A] n(Y) - - ( s ) dv nA Y
=
ty(S),
(11.5)
dP x (y)
=
h(y)
(ii.6)
dv x~ (cf. of
(ll.4). y
and
The functions s
and of
Let us n o t e foliation
q,
y
Pxs'q • C~(x)
i.e.
2
p ~(y)
is also a s m o o t h m a p p i n g .
dv ~ x d(v~o rl )) Px~ The f u n c t i o n
T
Moreover, (with
~ = g)
Ity(S)
for
for
for
-
ii
y (C~(x)
ITs(Y)
(y)
=
and
h
are
÷ C~(s)
functions
= s.
As
q
is a s m o o t h
i n d u c e d by the
foliation,
(11.7)
is a m e a s u r a b l e there exists
Pxs
T h u s one can d e f i n e
T~(y) .
f u n c t i o n of
a positive
s
function
and g
y. satisfying
(ii.i)
such t h a t
and
s (Cq(y)
-< g(B)
(11.8)
N
[A],
that
(11.9)
and t h a t
- iI -< g(B)
y (C~(x)
measurable
the P o i n c a r 4 m a p
- 1 I < g(B)
y 6 C~(x)
lh(y)
t
respectively.
and
s 6 A.
(ii.io)
135
Indeed, tion Ii.i.
(11.8) (11.9)
follows i m m e d i a t e l y from follows from the fact that
and from the C o n d i t i o n same as above,
(11.2) and from P r o p o s i -
(Bc) of Sec.
i.i I.
~
is a smooth f o l i a t i o n
(11.10)
follows from the
together with the T h e o r e m 7.1.3 I.
As one admits that C o n t i n u i t y Theorem,
q(B)
for any
= e(B),
then in virtue of A b s o l u t e
y E C~(x)
and
s 6 A
one can define
d(, DA Vy OPxy) (s) = H (s). d~ A Y x The f u n c t i o n (4.5)
H
(ii.ii)
is a m e a s u r a b l e
function of
y
and
s
and from
one deduces that
IHy(S) where
L
(11.12)
- 1 I ~ L(B), is a function s a t i s f y i n g
F i n a l l y let us defin~ formula:
if
D
(ii.i)
(with
~ = L).
the p r o b a b i l i t y m e a s u r e
is a m e a s u r a b l e
subset of
A,
~
on
A
by the
then
(D) = ~[A] ([D]) , where
[D]
is d e f i n e d like
implies that the m e a s u r e
~
[A].
A b s o l u t e C o n t i n u i t y T h e o r e m easily
is e q u i v a l e n t to the m e a s u r e
~A. x
Let
us define dgn A x
(~)
=
B(s).
(11.13)
d~ From
(4.5) and c o n d i t i o n
6 A,
the function
18(s)
where
f
Step 2.
- II
B
(Bc) of Sec.
5 f(B),
is a function satisfying Let
1.11 one deduces that for
satisfies the i n e q u a l i t y
Q c [A]
(ii.i)
(with
be an a r b i t r a r y m e a s u r a b l e
~ = f). subset.
It follows
from the u n i q u e n e s s of the set of c o n d i t i o n a l m e a s u r e s that to prove the T h e o r e m ii.i it is s u f f i c i e n t to see that v[A] ( Q ) =
I
[[ [A]
where
1
IQ R C~(a) (r)Ga(r)d~a(r)]d~[A] (a)'
(11.14)
C~ (a)
denote the c h a r a c t e r i s t i c Z the f u n c t i o n G a is such that
function of the set
Z
and where
136
IGa(r) - II < ~(B), for
~a
almost all
(11.15)
r 6 [A] N V(a).
From the definition of conditional measures
[A] ) } , from t~Ch(y
11.4)-(11.7) , (ii.ii), (11.13) and from the Fubini Theorem, one obtains that f
[A] (Q) =
II
IQ n C n (a)(s)d~[A] (s)~d~ [A] (a) ~ ~a)
[A]
C (a)
[A]
C (y)IQ n C (y)
c~ tY)
[A]
Cn (Y)IQ n C D(y)
q~YJ
Fr
IQ
x
(s) ty(S) d~y A (s)~ h(y) d~x (y)
[A]IQ (y,s)tY (S) h (y) dgyA (S) d~x (y) = (11.16).
[A]IQ(y'S)ty(Pxy(S))HY(~)h(y)dm~xA(s)d~x(Y)
=
(s) IQRC~(s) (y' s) ty (Pxy (~)) Hy (s) h (y) dm~x (y)~ dV~xA(s) =
(s)IQ n C~ (s) (y,S) ty(Pxy(S))Hy(S)h(y) • Tg (y) d ~ (s)IQ n C~ (s) (Y'S) [(y's)d~(y~d~Dx A(s) ([) IQDC~ (s) (y' ~) i (y,s) 8 (s) dv~ (y~ d~ [A] (y,s) , wi~h L(t,s) = ty(Pxy(S)) Hy(3) h (y) T~(y) . The fact that the last integral is equal to ~[A] (Q) is nothing else but (11.14) written in a slightly different manner. (11.15) follows now from (11.8)-(11.10), (11.12), (11.13) and from the counterpart of inequality (9.5) for five multiples. •
137
11.4.
From
COROLLARY
(ii.16)
ll.l.
one
For
immediately
~
almost
deduces
the
all p o i n t s
following
y 6 A
and e v e r y
point
X ^
z ~ A~ k,r,s,e,y
N U(x,q(B))
Ve~(z) (C~(z)) 11.5.
If
induced
= ~C~ (z) (V(z)
z 6 A£ k,r,s,e,y
we w i l l
Secs.
by
metric
suppose
11 and
that
V(z)
N U(x,q(B)) then
by t h e R i e m a n n i a n
Now, (cf.
such
that
v p
is e q u i v a l e n t
U
AZ
one
has
= i. we w i l l
z
on local
the
4)
N A = y
denote
stable
~-invariant
the m e a s u r e
manifold
V(z).
probability
to the m e a s u r e
9.
measure
Let us r e c a l l
that =
Ak'r's'e
I>2
k,r,s,e,7"
y>0 Theorem COROLLARY
ii.i
one
the
following
I_~f ~ ( A k , r , s , e)
11.2.
z E AZ k,r,s,e,y
implies
where
£
> 0
is big
then
enough
for
and
~
almost
y > 0
every
is small
point
enough,
has ^
~z(V(z)
Proof.
N Ak,r,s, e N U(z,q(B))
Let
G = M\H. measure
H
denote
As the ~,
For
~
then
the
also
G1 =
measure
= ~(G)
z ~ AZ k,r,s,e,7
us d e f i n e
set of r e g u l a r
invariant ~(G)
= 9z(V(z)
N U(z,q(B)).
points
~
(cf.
(11.17)
Sec.
is e q u i v a l e n t
2.2 I) and
to the
= 0.
let us d e f i n e
A
z
= {y ~ V(z) ; y ~ G}.
U {z 6 A Z - ~z(Az) Z>2 k,r,s,~,y'
> 0}.
G1
Let
is
y>0 measurable. Let such
us
that
First
we w i l l
suppose
that
prove ~(G I)
~(G 1 N A £k,r,s,~,7)
(with r e s p e c t As we
to
~)
suppose
one
sees
that
one
has
Vz(V(z)
z,
9z(V(z)
of the
that
on the
> 0.
> 0. set
v(G I)
set of
that
Thus Let
> 0,
> 0
x
one
can
find
be a p o i n t
£
and
y
of d e n s i t y
G1 N A~ k,r,s,~,y" then
z E U(x,q(B))
N G 1 N U(x,q(B)))
N G N U(z,q(B)))
~(G I) = 0.
> 0.
in v i r t u e
of T h e o r e m
of p o s i t i v e This
implies
and c o n s e q u e n t l y
ii.i
~
measure
that
for s u c h
from T h e o r e m
ii.i
138
one obtains Now, and
from
that
~(G)
l e t us n o t e
INFINITE
12.1.
1-3 of
infinite
and
[Rue] 3) a n d
b y M. B r i n
and
4.1.
ik,r,s,
indicate
From
is t r u e
that
v ( G !) = 0.
(7.2.1) I,
for all
\G 2 c H.
how, our
counterpart
The last
Z. N i t e c k i
•
be very
using
some
previous
of T h e o r e m s
result
(see
a n d P. C o l l e t
We will
~ ( G 2) = 0.
(11.17)
can modify
dimensional
F. L e d r a p p i e r subject.
one
proves
CASE
now quickly
the
that
because
[Rue]3,
contradiction
G 2 = G @ GI;
DIMENSIONAL
We will
Secs.
This
(7.2.2) I it f o l l o w s
z ( ik,r,s,~\G2,
12.
> 0.
has
[Bri]2).
been We
to o b t a i n
6.11
[Man] 2
thank
and we leave
from
arguments (cf.
proved
for t h e d i s c u s s i o n s
sketchy
results
independently
sincerely
M.
we had about the d e t a i l s
Brin,
the
to t h e
reader. 12.2.
Before
recalling
the existence emphasize
that
the a p p r o a c h approach
of LSM even
of
(cf.
in f i n i t e
21 a n d
purpose
only
of P a r t
for t h e
Thus
the m a p p i n g s
Indeed,
invertibility
12.3.
E
be a separable,
be an
open
Let us suppose
that
K.
that at least
The well one
the
u p to now,
Indeed,
K c U
known
such measure
of P a r t s
to -~.
the
of
metric
of
and, whose
for
the
}: N ÷ %(N) of
in w h a t
%
is u s e d
follows,
existence
is
~.
Let
dimensional
¢: U ÷ E
is a c o m p a c t
n0 > 0 be a
I-III we can
Moreover,
invertibility
metric
infinite
subset.
#(K) c K, a n d t h a t for s o m e no d¢ (x) is c o m p a c t . Let ~ on
equal
of L y a p u n o v
the
measure
allows
let us
to the P e s i n
2).
of i n v e r t i b i l i t y I,
a k i n d of L y a p u n o v
of
Let
LCE
in P a r t
construction
construct
U c E
compared
for t h e p u r p o s e s
with
independent
and
space,
considered
from Appendix
I, the a s s u m p t i o n
c a n be d r o p p e d .
we will
case
advantageous
(T.I)
[Rue] 3) w h i c h
Hilbert
+~
is in f a c t u n n e c e s s a r y . also
(cf.
(1.2) I)
I log+fld~l[ld~< consider
framework dimensional
dimensional
[Rue] 3 is m o r e
(cf. Sec.
condition
Ruelle's
in i n f i n i t e
and
subset
every
%-invariant
Krylov-Bogoliubov exists.
Hilbert
be a
C1 such
x 6 K, Borel
space
mapping. that the o p e r a t o r
probability
theorem
guarantees
139
L e t us s u p p o s e
I
that
log+lld~xlld~j(x)
(12 .i)
< +~.
K
For
every
x (x,u)
The
x ( K
=
1-3 of case
in A p p e n d i x
2.
in a s o m e w h a t
theorem,
the O s e l e d e c Let
us n o t e
different
the LCE w i l l
let us d e f i n e
the LCE
X(x,u)
all
need.
the n e c e s s a r y
It g e n e r a l i z e s
Multiplicative
that
manner
in
unlike
the r e s t of
are
formulated
formulated
the book. way;
proved
infinite
Theorem
facts
in d e c r e a s i n g
facts
to the
Ergodic
[Rue] 3 t h e s e
be e n u m e r a t e d
by
(12.2)
summarizes
[Rue] 3 that we
dimensional
section,
u 6 E
1 logIId~n (u)If-
lira sup n++~
following
in Secs.
and
In this
i.e.
Xi(X)
>
Xi+l (x)THEOREM #(A)
12.1.
c A,
(12.1.1)
There
p(A)
= 1
The
exact
and
for e v e r y
u
6 E.
When
u
varies
number
of d i s t i n c t
Xl(X) where
> X2(X)
s(x) = + ~
determined of
E, of
exists
limit
in
that
in
subset
A
for e v e r y
(12.2),
E\{0},
> .....
is not
finite
such
values
sequence
a Borel
x
perhaps
k(x,u)
{Xi(X)}
of
takes
includinq
K
such
that
E A:
equal
too
at m o s t
-~,
exists
a countable
-~,
> Xs(x) (x) = -~,
excluded.
of c l o s e d
codimension
Moreover,
linear if
there
subspaces
exists
a uniquely
{Li(X)}l~iss(x)
i < +~, def
E = Ll(X) such
that
X(X,U) (12.1.2) closed
V
~ L2(x)
for e v e r y
L e t us linear
fix
and
for e v e r y
c A,
T < 0.
subspace
= {u 6 E;
¢(A)
i
~ Ls(x) (x)
~ Ls(x)+l(X)
{0}
u (Li(x)\Li+l(X)
o n e has
= Xi(X).
n As
~ .....
this
lim m÷+~ exact
For
every
V n = Vn(X)
of
n ~ 0 E
let us d e f i n e
by
ull = X ( ~ n ( x ) ,u) ! logIId¢ m m ~n(x ) limit
exists
the
in v i r t u e
of
-< T}
(12.1.1).
(12.3)
140
Then
for e v e r y
has
d~n(L) x
(12.1.3) we
linear
~ V
For
n
subspace
every
a > 0
A
such that
defined
on
IIdCmn
vll ~
L c E
satisfying
L ~ V0(x)
one
= E,
(x) = E. there
exists
for e v e r y
a positive n , m ~ 0"
and
measurable every
v
function E Vn(X)
IlvlI~ (x) e x p (m T) e x p ( ( n + m ) e).
(x)
(12.1.4)
If
L
is a l i n e a r
subspace
of
E
such
that
L N V0(x)
= {0},
then lim 1 log y(d~(L)
where
the angle
y(-,-)
L e t us n o t e Vn(X) ,
n ~ 0,
for e v e r y
,Vn(X))
that are
n ~ 0,
was
=
defined
(12.1.1)
L e t us n o t e
all o f the
12.4. Sec.
also
f r o m Sec.
We will
(2.15).
(12.1.2)
same
finite
imply
that
codimension
the
subspaces
in
E.
Moreover
2.2
from
is a n i s o m o r p h i s m .
that
(12.1.1)
follows
are directly
3.1 o f
now describe
from Corollary
related
to the c o n d i t i o n s
[Rue] 3 r e s p e c t i v e l y . in o u r
framework
the c o u n t e r p a r t
of
2.3 I
L e t us
fix
T < 0.
A T = {x E A; X £ ( x ) (x)
Obviously the
by
X
(12.1.2)-(12.1.4)
(S.2)-(S.4)
and
de n : L + d~n(L) X
[Rue]3;
0,
AT
introduced For
I
f(x)
(x)
x
L e t us d e f i n e
> T >_ X z ( x ) + l ( X )
is m e a s u r a b l e notation
6 AT
and
Vn(X)
let us d e f i n e
= exp XZ(x)+l(X),
e x p X £ ( x ) (x).
the
¢(AT)
set for
c AT .
= Lz(x)+l(~n(x))
some
Z(x)
L e t us n o t e
>_ i}.
that
for x ~ A T.
in
141
Clearly
l(x)
Let
r
us d e f i n e 0 < e ~ £
< 1. and
s
£r,s
be
positive
exactly
as
in Sec.
let us d e f i n e
r,s
the
l(x) A
This
> 0.
r,s,e
last
reader
A
=
case
r,s
Now,
l'(x)
For
in
positive for
x
8e
Now, n , m ~ 0;
in
the E
now
has
in
leave
it to the
of the
it f o l l o w s
dimensional subspace
for e v e r y
(12.6)
subspace
V0(x)
m,n ~ 0
W0(x)
(cf.
(12.3)).
and e v e r y
(12.7)
,
function
that
there
AT,
such
defined exists that
on Ar,s, e. a measurable
(cf.
(2.1.3) I)
(12.8)
~ Y(Wn(X),Vn(X)),
< +~.
"
Now,
For
every
of the
the
on
~' (x) = exp(-£).
(2.1.i) I)
measurable
d~(W0(x))
E
Moreover,
c Ar,s,s.
n t 0
let us r e m a r k indeed
(12.5)
for x ( Ar,s,e:
finite
that
(cf.
defined
every
d i m Wn(X) ; p(x)
codimension
one
(12.1.4)
W n ( X ) def
and we
~' (x) = ~ ( x ) e x p ( - 3 e ) ,
positive
and
here
~(hr.~.s)
IIVlI~s(x)rmexp(s(n+m))
function E AT
e,
hr,s, £
it f o l l o w s
(Be(x))-lexp(-en) where
some
complement
(12.1.3)
from
Let
S r < s S ~(x)}.
and
let us d e f i n e
2£ is some
Moreover,
=
to
E Vn(X) , x ( hr,s,c,
where
I"
r < s. For every
< r < s < ~(x)}
(2.6)i-(2.8) I we d e f i n e
x E AT,
lld%~nxVll _
0 there
exists
such t h a t for all
a measurable n,m ~ 0
positive
and e v e r y
(x)
n
1 lld¢~n vll >-Nvll y - ~ [ ~ ( x )
]mexp(-s(n+m)) .
Let us fix once and for all a n a t u r a l Ap =
{x ~ AT; p(x)
number
(12.9)
p _~ Z. Let us d e n o t e
= p}.
It is c l e a r
that ~(A ) c A . P P Let x E A . L e t us d e n o t e by ll(X) ~= 12(x) ~= ... all L y a p u n o v P c h a r a c t e r i s t i c e x p o n e n t s at x , w h e r e a ny e x p o n e n t Xi(X) is r e p e a t e d k.l(x)
times
following base
; i.e.
k2(x)
the first kl(X)
with
fl(x) ..... fp(X)
i,l _-< i _-< p l i m n÷~
1 n
9 x
log
every
Ep
vectors
1 -~ and
{fi(x) }iEp
¥(d~n(Ep),
(12.12)
n ~ 0 P Z i=l
Xl(X) , the
an a r b i t r a r y
such that for e v e r y
= U(Ap)
subsets
implies
C
a ~ invariant
such that for e v e r y x E Z and for
p,Q
c {1,2 .... ,p } we have:
1 (x)
exp
c that for
al,...,ap
(12.11
subspaces
of W 0 (x)
spanned
by the
respectively.
there exists
n
d~x(Eo))~= -
mapping
d,n(EO) ) = 0
{fi (x) }i~Q
e > 0
to the m e a s u r a b l e
that for p ~_ 2 there e x i s t s
on Z such t h a t for e v e r y
easily
let us c o n s i d e r
2 applied
the l i n e a r
and
and for e v e r y
laiI
disjoint
EQdenote
> 1 defined
Now,
u(Z)
log y ( d , n ( E p ) ,
T h u s for e v e r y C
, implies
Z c Ap,
two n o n - e m p t y lim n÷~
where
subset
Now, W0(x),
with
n IId~x(fi(x))II = li(x)
of A p p e n d i x
> d*xiw0(x )
measurable
etc.
of s u b s p a c e
The T h e o r e m T.I. Ap
X2(x),
of t h e m c o i n c i d e s
a measurable x E Z
positive
and e v e r y
(-
n)
u-almost
every
function
n ~ 0 (12.12)
x ( Z, for e v e r y
E IR
n n ~ IId~ (fi(x))II =< H e (x) exp(en) II d ~ x ( i = 1
a .1f . 1 (x)) II (12.13)
143
where
qe(x)
implies
=
(4C
(x)) p.
t h a t for e v e r y
1 y ( g l U , e 2 v) -~ ~,
Indeed,
two v e c t o r
81 = ±i,
the s i m p l e
geometrical
u, u ~ E,
u ~ 0, v ~ 0, such t h a t
g2 = -+i, one has
consideration
IIul[ + IIvl[ _- 0
converges.
the norm
verifies
II'II ~ II'II~,x ~ A exp( n6)rl-[i
(12.16~
145
and the first Indeed, obtains
that
inequality
of
from
(12.15),
for
u 6 V
n
(12.2.2). from Schwarz
inequality
and from
(12.7)
one
(x)
r IiUjix,n
= /
[ (~' (x)) m>_0
-2mltd~ ~
(u) II2 ' nx
_
0
-2(m+l)ild~m~l (u) ll2' (x)
l' ( x ) / ~ (I' (x))-2mIld~m
m~0
We pass on subspace define
now to the c o n s t r u c t i o n
of the
Wn(X).
u 6 Wn(X),
For any vector
!
(u) ll2' = I' (x) IiUIix,n-
~n(x)
scalar
product
u # 0,
_0 lld#m
(u) II
0,
'xtn
~ aibi , i=l
i.e. with respect normal
and
product
!
II'Nx,n -0 [Id~nx(fn, i) II
[ exp(-2~m) m>_0
]Id~nfill
_
Xl
(x) n ,
are
finally
one
w i t h B = -2-
Let us go n o w to the r i g h t x 6 ~l p,r,s'
v 6 Wn(X).
llull-< IIUIIx,n.
(12.19),
orthogonal
u ~ Vn(X),
!
exp(-~n)
Now
w = u + v,
side i n e q u a l i t y
and
of
(12.21).
As
(12.10),
1 Y ( V n ( X ), W n ( X )) >_ ~ e x p ( - 6 n ) . The s i m p l e g e o m e t r i c and
v 6 W
n
considerations
imply
that for every
u 6 Vn(X)
(x)
I]u]l + ]]vll -< 4Z e x p ( ~ n ) llu+vll. consequently,
from
(12.16)
and
(12.19)
one o b t a i n s
llu+VIlx,n _< IIUlIx,n + IIVllx,n _ < E exp(dn) llull + Z e x p ( n ~ )
NvH
0
In fact
enter
12
mapping
and
scalar
that besides
constant
lld~xlI ~ L
the
the with
in the
Moreover
in a b s e n c e
indices.
the e s t i m a t i o n
(6.1.5)
is
I
because
G (x) A 2 (x)
< ~.
x~ 1 p,r,s In the o t h e r TLSM 12.7. such
words,
are u n i f o r m l y N o w we w i l l that
A1 k,r,s,e,y
for
the m a p p i n g s
@x
which
defines
Lispchitz. define
~(AP'r's\/>0U one
x 6 ~l p,r,s'
the m e a s u r a b l e
A P1 ' r ' s ) = 0
considers
the
sets
and
subsets that
A1 p,r,s'
A1 p,r,s
if i n s t e a d
c ~l p,r,s of the
the C o r o l l a r y
7.11
sets
remains
150
valid. The space of all space
is separable if
p
dimensional
p < ~.
subspaces of a separable H i l b e r t
The space of all scalar products on
finite d i m e n s i o n a l vector space is also separable. theorem
(see e x e r c i s e II. 7.3
du calcul des probabilit4s, measurable that
W0(x)
it depends c o n t i n u o u s l y on
As
Neveu, Bases m a t h e m a t i q u e s 1964)
such that
one can find a
~(~lp,r,s \A~ ~,r ,s ) ~ ~'
and the scalar product x 6 AZ p,r,s
_ 0
i > 0,
to the m e a s u r e
for
if
Jl # J2'
1
-
1
f (A~ i ) =
1,
A~
such that
and
1
the
mapping
~ IAi
has a
n.
l
COROLLARY
13.1.
continuous Ai
is a
authomorphism.
If for some
spectrum K
i > 0,
(see Sec.
2 of
the m a p p i n g
[Roc] 3 ) , then
$ restricted
to
authomorphism.
In fact t h e r e
is no d o u b t
t h a t the m a p p i n g
~ni A~
is B e r n o u l l i
l
(cf. T h e o r e m
8.1 f r o m
n e e d an e s t i m a t i o n given
in T h e o r e m
13.3. from
[Pes]3).
of J a c o b i a n
J(p),
t h a t to p r o v e
stronger
this r e s u l t we
t h a t the e s t i m a t i o n
4.3.
Let us f o r m u l a t e [Kat] 2
It seems
(Theorems
now the c o u n t e r p a r t
(4..1)-(4.3))
of the p r i n c i p a l
w h i c h do not i n v o l v e
results
topological
entropy.
Sec.
Let
M,N,~,#
1 I.
Let us d e n o t e
of p e r i o d THEOREM Lyapunov
13.2.
satisfy by
conditions Pn(~)
(A),
(B) and
(1.1)-(1.3)
the n u m b e r of p e r i o d i c
points
n. Let us s u p p o s e
exponents.
Then
t h a t the m e a s u r e
U
has n o n z e r o
from of
156
(13.2.1)
periodic
points
(13.2.2)
max
(13.2.3)
if in addition
of
~
are dense in the support of measure
P, O, lim sup n+m
log Pn(~) ) h (~) n > '
the measure
concentrated
on a single trajectory,
points o_~f ~
having a transversal
support of measure
p.
p
is ergodic and not
then the periodic
homoclinic
hyperbolic
point are dense
i__nnthe
PART III THE E S T I M A T I O N OF E N T R O P Y F R O M B E L O W THROUGH L Y A P U N O V C H A R A C T E R I S T I C E X P O N E N T S * F. L e d r a p p i e r and J.-M. S t r e l c y n
i.
I N T R O D U C T I O N AND F O r m U L A T I O N OF THE RESULTS
i.i.
The c e l e b r a t e d Pesin entropy formula asserts that when
compact R i e m a n n i a n manifold, when an a b s o l u t e l y c o n t i n u o u s the m e a s u r e on
M
~ ( Diffl+e (M)
and when
is a is
(i.e. a b s o l u t e l y continuous w i t h respect to
induced by the R i e m a n n i a n metric)
p r o b a b i l i t y measure,
M ~
invariant
then
r =
(
h~ (~)
where
]M
Xl(X)
< X2(x)
exponents of
#
respectively. system
[ ki (x)xi (x)) d~ (x) , Xi(X)>0
at
2
y>O O~ksm (cf.
Secs.
(6.5) I,
3.5 I)
(2.8)ii
We call
where and
i+
thus
the
A+
=
Let
us d e n o t e
=
0 < e < i, of P a r t s
results
following
{x 6 A ; X + ( x , v )
A+ r,s,e
e = e(r,s), all
> 0
subset for
of
some
satisfies
I and
(2.5) I,
II are
applicable.
A: 0 ~ v
( T N}. x
also
A+ N A
and
P+
=
U r0
=
(1.3)
lim ~n l°gN (d~)All n÷~
and i ( x~ ki(x)Xi(x))d~(x) M X. ( )>0 1 Thus,
=
lira 1 i l°glI(d@n)AIId~(x)" n÷~ n M
the Pesin entropy formula may be also w r i t t e n as follows:
h
1
:
r
(1.4)
lira n I l°gtIId l lld (x)" n÷~
M
In their i m p o r t a n t work
[Led]6, F. L e d r a p p i e r and L. S. Young
prove that for the d i f f e o m o r p h i s m s of class manifolds,
the p r o p e r t y
C2
of smooth c o m p a c t
(1.4), or e q u i v a l e n t l y the e q u a l i t y in
is a c h a r a c t e r i s t i c p r o p e r t y of Sinai measures.
(1.2)
Earlier L e d r a p p i e r
(Led]4,5 proved this fact for m e a s u r e s w i t h n o n - z e r o L y a p u n o v exponents. It seems that the proofs from maps w i t h s i n g u l a r i t i e s
[Led]4, 5 can be adapted to the case of
studied in this book.
Let us note that i n e q u a l i t y mappings
formed by all finite c o m p o s i t i o n
s a t i s f y i n g the c o n d i t i o n s Secs.
2. 2.1.
(1.2) remains true for the class of
1.21 and 4.2i) .
(A),
(B) and
~K o...o ~i
(2.1)-(2.3)
of m a p p i n g s
from Sec.
1 I.
(See
We leave the details to the reader.
PRELIMINARIES If
H
is a finite d i m e n s i o n a l E u c l i d e a n space, we denote
the volume on dimension, mapping.
H.
E1 c E
Let
E
=
F
VOlEl(U)
Vol
H
be two E u c l i d e a n spaces of the same
be a linear subspace of
Let us define VOlFI(A(U))
IAIEll
and
E
and
A :E ÷ F
a linear
163
where
U
arbitrary A(U)
is an a r b i t r a r y
open
linear
of
c F I.
We
subspace
also
and b o u n d e d F
of the
subset
same
of
E1 , F1
dimension
as
is an
E1
and
denote
Idet A I = IAIE 1 . Let
X
dimension
Y
JX
induced
be d e n o t e d
2.1.
~ x formula.
If
(M,M,~)
a measurable
and
~
by the R i e m a n n i a n
metrics)
respectively.
We r e c a l l
y
f E Ll(y,Vy),
be a m e a s u r e
measure
on
X
the
then
r Iy fd~y.
=
(f o T) Idet dT1d~ x
Let
known
the m e a s u r e s
of v a r i a b l e s
PROPOSITION
2.2.
be two R i e m a n n i a n m a n i f o l d s of the same f i n i t e 1 T :X + Y be a C diffeomorphism. Riemannian
(i.e. will
change
Y
and
measures and
and
preserving
space map.
of We
finite shall
measure
use
the
and
T :M~M
following
well-
result.
PROPOSITION
2.2.
defined
M
on
Let
such
g
be a p o s i t i v e
finite
measurable
function
that
log - ~ o T E LI(M,~) g
,
where
log - a
=
min(log
a,0).
Then lim ~ log g(Tnx) n
f
=
log g ° T d ~ g
Proof.
Let
immediate to the
us
and
=
first (2.1)
0
~-almost
everywhere,
(2.1)
(2.2)
0.
note
that when
follows
from
log g E L I ( M , ~ ) ~
the B i r k h o f f
ergodic
(2.2)
is
theorem
applied
function
log ~ ° T. g Let when h
us also
applied
E LI(M,~),
note
that
to a f u n c t i o n but
the B i r k h o f f
ergodic
h, h = h+ - h_,
in g e n e r a l
the
limit
theorem
with
is still
h+ ~ 0, h_
can be i n f i n i t e .
As
true
~ 0
and
164
log - 9. ° T 6 L 1 (M ,~ ) , g this
shows
that
n-i lim ~ [ n+= n i=0
the
following
Ti+l log g o g o Ti
limit
exists
~
almost
everywhere
1 Tn l i m -- log g o dsf K n+~ n g
=
and moreover I
Kd~
f J
=
M
where
log ° ~ g T
d~,
M
both
sides may be equal
+~.
As 1 -- l o g n we have K
g
÷
almost
0
everywhere
therefore =
l i m ! l o g ( g o T n) n
On the other
hand,
almost
w e know,
as
everywhere.
almost
0 < g
0r k i ( x ) X i ( x ) ) d u (x)
Proof. in
The p r o o f
=
is b a s e d on the f o l l o w i n g
lOse] b u t e x p l i c i t l y
in
(2.3)
fM log T U ( x ) d U (x).
fact i m p l i c i t l y
[Rue] 1 (compare w i t h
(1.3)).
contained
If
x ( A
then
Ix
k. (x)X, (x)
X. ( )>0 1
1
=
1
lim 1 l o g IdSxn I ul "
n÷~ n
(2.4)
E
x
F r o m the H a d a m a r d
inequality
1
we h a v e
1
-1
< Ild¢¢(x)N m - ]d$$ x) I u
I
-
0 A Z.
and
x ( AZ
This
is p o s s i b l e
and t h e n F o r any
defined
such that
x r,
3.1 in a p a r t i c u l a r ~(S(x,r))
by c h o o s i n g
> 0
first
case.
for all £
We
r,
such t h a t
in the s u p p o r t of the t r a c e of the m e a s u r e 0 < r 5 r£,
by all the sets
to
N B(x,r)
A£ N B ( x , g ( £ ) r ) ,
from results
and
,~j t-V 3
~) '
~j = ~ , £
n t 0
the two p o i n t s
If one t a k e s as
m-i £+i Aj U U k=O j=2 k , ~ , l +
and
pV(y,z);
U y ( A £ N B (x,e (£)r)
belonging
This proposition
B ( x , r Z)
-nC£
) ~ B£e
and are not in the same l o c a l y
from
of
the c l o s e d ball of c e n t r e
i__nn V(y)
r,0 < r ~ r£,
S(x,r)
some
N
topology);
then for e v e r y
-n z
y,$
is c o n t i n u o u s
into the s p a c e of s u b s e t s
(endowed w i t h the H a u s d o r f f
y
Vloc(Y)
is c o n n e c t e d ;
B(x,g (£)r£) N i£
(3.3.4)
is such that
we c o n s i d e r
the p a r t i t i o n
~r
171
Vloc(Y) for
Y
D B(x,r)
E A Z D B(x,s(£)r)
follows
clearly
that
and the set
~r n
We d e f i n e a partition
0r = n = 0 ~r
M\S(x,r).
is a m e a s u r a b l e ~r"
for some
The p a r t i t i o n
r,
From
partition
0 < r ~ rZ
n
(3.3.3)
of
it
M.
of L e m m a
that we c h o o s e
3.1 w i l l be later.
Let
us d e f i n e S
=
U nt0
r
%ns(x,r).
We n o w p r o v e p r o p e r t i e s
(3.1.1)
This property
(3.1.2)
It is c l e a r
C
(3.1.1)-(3.1.5)
is c l e a r
~(S
f r o m the d e f i n i t i o n
that for
(z) c C n V l o c ( ~ - n z )
when
z ( S
r
r
) = i.
of
a n d for some
H r.
n > 0.
c V(z) .
~r On the o t h e r ~r'
8r ~ 0,
hand,
we c l a i m t h a t there
such that
y (V(z)
pV(y,z)
exists
a function
} Sr(Z)
implies
y 6 C
'
The p r o o f of choosing
r
consists
in c o n s t r u c t i n g
such that
We d e f i n e
E(z)
(3.1.2)
=
8r
(z). ~r
8 > 0 ~ r o n l y on U A~.
such a
B
r
a n d then
almost everywhere. For
z ~ U A Z, Z
put
inf{£' ;z E A z , }
and nC£ Br(Z)
=
inf{A£ 1 n>_0 (z) ' 2Bz(z)
p (¢-nz,3B (x,r)) e
L e t us f i r s t p r o v e our claim. pV(y,z)
S 8r(Z).
C ~r(# -n y )
=
We h a v e to c h e c k
that
y
6 Vloc(Z)
z 6 U AZ,
that for any
y
i }r . ' BZ(z)
E V(z)
and
n ~ 0
C~r(~-nz)
F i r s t we k n o w by
any
Let
(z)
(3.1)
(3.3.4)
and that
as
y
(3.3.5)
6 V(z)
and
applies.
pV(y,z)
~ AE(z)
T h e r e f o r e we h a v e
for
n ~ 0 -nCz pv(~-ny,~-nz)
< Bz(z)e
(Z)Pv(Y,Z)
i < ~ P(#-nz, ~B(x,r))
(3.2)
172
and -n pv(~
We h a v e (i)
-n
by
z) pV(y,z)
z) < B£ (z)e
four c a s e s
If
(ii)
-nC~
-n y,~
y
to c o n s i d e r . ~ -n
and
(3.3.6)
and
If n e i t h e r
z
both belong
If
versa,
~-n y
~ -n z
nor
~-ny
belongs
-n
y,~ (x,r))
which would proves
we h a v e
(3.1)
belong
of
~r"
to
S(x,r)
to
S(x,r),
but not
we have
~-nz,
or v i c e
< pv(~
contradict
-n
y,{,
(3.2).
z)
Thus only
(i) and
(ii) occur,
which
the claim.
We w i l l everywhere. of
S(x,r)
we s h o u l d h a v e
-n
pv(¢
to
(3.3).
(3.1) by the d e f i n i t i o n (iii)-(iv)
(3.3)
_< r.
r, 0 < r 0 Let
x ( M.
[0,rz]
defined
v(A)
=
and let
p
applied
to
Let
~
~
almost
8r > 0
such that
for L e b e s g u e
almost
~
almost
every
choice
everywhere.
be the f i n i t e n o n - n e g a t i v e
measure
on
by
p({y (M;p(x,y)
6A}).
be an i n t e g e r , -Cp a = e , that
p >_ 1. IKpl
We g e t by P r o p o s i t i o n
= r,
3.2,
where -kC
K
=
{r;0 _< r < r£,
[ ~({y (M;Ip(x,y)-r I <e k=0
P As
}
preserves
the m e a s u r e
~,
P]) < + ~ } .
we have also -kC
K
=
{r;O __ Sk+ I.
Thus
l i m su~ = l i m k+~ k~ =
lim
- H(~
v...v
it is e a s y from
(2.3)
s I +...+ k
sk
H(~vT-I~
v...v k
to see one
that
for e v e r y
has
T-k~)-H(~) =
h(T,~)
.
k >_ 1
one
183
As
s I { s k ~ Sk+ I, P r o p o s i t i o n
following
Corollary
COROLLARY
2.1.
which
For e v e r y
2.2 i m p l i e s
is at the b a s i s finite
immediately
the
of the p r o o f of T h e o r e m
measurable
partition
~
of
M
i.i. one
has h(T,~)
3.
CONSTRUCTION
3.1 in
~ H(T-I~I~)
For N
x 6 N
OF P A R T I T I O N S and
of r a d i u s
Let us n o t e h o o d of
r > 0
r
{~t}t>l
by
and c e n t e r
N(r)
B(x,r) at
= N~Ur(A),
one d e n o t e s
the c l o s e d b a l l
x.
where
Ur(A)
is the o p e n r - n e i g h b o r -
exists
a finite measurable
A.
PROPOSITION partition
3.1. Pr =
F o r any
r > 0
there
(C0'Cl'''''Cp(r))
of the s p a c e
M
such that
A c C O c U2r(A )
and for all
i,
C i n Ur(A)
(3.1.1)
1 ~ i ~ p(r),
: @,
(3.1.2)
d i a m C. ~ 2r
(3.1.3)
1
Ci
Proof.
contains
As
an
N(r)
open
ball
in
N
of
r
radius
~.
(3.1.4)
is c o m p a c t , one can find its f i n i t e c o v e r i n g by r ~, c e n t e r e d at p o i n t s of N(r). Let r = B ( Z ~ ( r ) , ~) be a c o v e r i n g of N(r) of
c l o s e d b a l l s of r a d i u s r B 1 = Bl(Zl,~) , .... BZ(r) minimal
cardinality
One can a s s u m e BI,...,B s
t h a t the b a l l s
are p a i r w i s e
s B~j n (k=l @ Bk)
~ 0.
a m o n g the b a l l s Now,
by such c l o s e d
disjoint
Let us d e n o t e
{Bi} s
1
}
"
(4.4)
188
Let
2
y E B(w,
d 2)
be an arbitrary point.
Let us denote by
tl-n~(n) F
the shortest
~(r)
geodesic
2
~
where
joininq ~(F)
w
and
denote
y; F c B(w,RN(w,N))
the length of
F.
and
Now, by Mean
tl-n~(n)d 2' Value Theorem, t ~ t0(n)
using condition
(1.4) I and
~ t~(n)
p(¢(w),¢(y))
_ tk(n),
0 < k 2t ~
(4.5) Let
y 6 ~k(s)
~k(s)
be an arbitrary point.
c B(w,R(w,N)).
Let us denote by
ing w and y, F c B(w,R(w,N)). 1 Z(F) < 2 (2dc3) k t I- (n+k) ~ (n) d" Now, by Mean Value Theorem, one obtains
that for
p(¢(w),¢(y))
II
sup zEF
1 e(n)d (2dc3)k tl_(n+k)
F
(4.1.3) k it follows
the shortest
As follows
from
from condition
t >_ tk+l(n)
S £(F)
From
geodesic
that join-
(4.5),
(1.4) I and from
(4.5)
I
> tk+l(n),
lld#zlI
tk+l " (n) -
~ H nt, then from
(4 1.2)k+l one obtains
that for
189
#k+l(s)
c {x E M;
p(x,A)
1 t l _ 2 n e (n) d } c
2 (2dc3 ) n t~(n)
c {x E M;
p(x,A)
> i } -
Indeed,
for
t
t~(n
)
•
1 te(n ) > (2dc3)n
big e n o u g h
1 tl_2n~(n) d •
Thus
(4.l.1)k+ 1 is proved. From
(4.1.3)k+ 1 z 6 ~k+l(s)
(4.1.2)k+ 1 and
it follows
the first
follows,
for
inclusion
of
inclusion
exists
(4.1.3)k+ 1 is proved. fact that
(4.1.3)k+ 1 follows
IV ~ tk+l(n)
tk+l(n)
(2dc3) k+l
of
!)
tl_(n+k+l)a(n) d + t
IV 1 >_ tk+ ,,,I, from the t ~ tk+
The third
1 tl-(n+k+l) e(n)d
such that
1
from
(4.1).
Indeed,
t ~ tk+l(n)
one has
C < min (CR, C tg~(n-------~tge(n-----~) -
_ 0, let us n o t e
"Main
remark
Lemma"
used
There
exists
a number
is in our
in the
framework
Ruelle's
E G(n,t)
it w i l l
variation
and
PROPOSITION
proof
the e x a c t (see r e m a r k
counter(a)
For such
that
such
that
for e v e r y
for e v e r y
t ~ t(n)
from
n > 2
and
for
x E S
~ ~} ~ KH (d¢~)hl I. from
following
5.1.
C 1 = Cl(m,a,r)
K > 0
such
every
be c l e a r
of the
a number
~(n)
E ~t; A n Cn(s)
As
(5.1)
[RUe]l).
LEMMA
#{A
and
T -1 o d~ n o T . %n(x ) X X
x
every
(m m, li.H))x( v.
÷
the proof,
this
geometrical
any
for e v e r y
lemma
is a n o n l i n e a r
fact.
m ~ l, a ~ 0
that
(5.2)
T
and
r > 0, t h e r e
E i(~m
, ~m)
exists
one has
V o I ( U a [ T ( B ( O , r ) ) ]) - ~ s. i=l 1
T(B(O,r)) for
V o I (_U a [ _ T(B(O,r))
]) 1
to the c o c y l e
of the c o c y c l e
at the p o i n t
integers
cocycle
g i v e n by the f o r m u l a
k odfxk(m) OTx I) f (m) x
the n o n - n e g a t i v e
can be a p p l i e d
tic E x p o n e n t s
GL(dim M,~)
+ (Tk(x)'T
fxk = fTk_ I (x) o . " °.f T ( x. ) ° f.x ' k
the m u l t i p l i c a t i v e
( Uj.
such that
~ > 0
such that
d ( m l , m 2) ~ B, t h e r e ml,m 2 ~ U i
The use of local
and
coordinates
in
Ui
197
and
U. gives the p o s s i b i l i t y to d e f i n e the n u m b e r s (the norms) 3 IIdfx(m I) - dfx(m2) ll if o n l y d ( m l , m 2) ~ ~; we l e a v e the d e t a i l s
to
the reader. T H E O R E M A.I defined every sup x6X i~£
Let us s u p p o s e on
[0,B]
ml,m 2 ( M
that there
such that
¢(0)
exists
a non-negative
= 0, l i m %(r) r÷0
= 0
function
and t h a t
o n e has
(A.4)
lldfx(m I) - dfx(m2) lI ~ # ~ d ( m l , m 2 ) )
d ( m l , m 2) 0_
of d i f f e r e n t i a b i l i t y was
,
is f i n i t e
the p o s i t i v e
t h a t all
z
1
{Ttv}
half-trajectory of t h e
=
flow
in of
Y
~. Z
a is
As
it
is a l w a y s
in t h i s p a p e r . t,
the m a p p i n g
at which
straight
line
Tt
may
it is c o n t i n u o u s . passing
through
piece.
the b i l l i a r d the m e t h o d
of
flow
{T t}
"section"
in its p h a s e which
is q u i t e
204
natural
for this
ergodic
properties
properties
problem.
F
section
used
the
is w i d e l y
F
could
section
what more ing t w o
3.2.
in Chap.
smooth
involved.
These
subsections
that
VI
of
by that
induced
descriptions
from the definition
of
smooth
closed
curves
of s m o o t h
closed
arcs
is n e v e r
contains
arcs.
Moreover,
Dividing
the closed
purely
of
particularities
arcs
of
rather
when
then we will
{Lj}j~ 1
class r (jUIF j)=
C k , then we will
Li
belong z ~ L.. l tangent of
and
a finite
Fj
follow-
under
arcs
number
these
curve,
or infinite
points curves
F =
then
number
of
( U L~) U ( U F ) i~l ~ j~l J eliminate
F i, w e c a n
of
F.
Further-
at all.
The
in o u r c o n s i d e r a t i o n s
given
to o n e of t h e s e
one depends
the billiards
only
with
that all curves
If the b o u n d a r y
of c l a s s
consider
of a t
a
repre-
o n the
consideration.
assume C k.
and
2).
closed
representation
plays
of a t
without
t w o of
is no u n i q u e n e s s
to a n o t h e r
we c o n s i d e r
of c l o s e d
F. a r e of c l a s s ] W e w i l l say t h a t
any
there
than
a r e of c l a s s
number
all
curves
...
...
1 and
representation
from this L
always
finite
with
the
of t h e p r o b l e m
In p a r t i c u l a r , boundary,
in the
r2,
Any
finite
role and the preference
F
of
is s o m e -
is a u n i o n
L2,
(see Figs.
the c l o s e d
curves
FI, LI,
ends.
either
1 (i~iLi) U (j~IFj)_
F =
secondary
sentations
point
F
a nondifferentiable
as a u n i o n
in the d e f i n i t i o n
representation
for t h e i r
one common
F
unique.
completely
perhaps
this curve
smooth closed
more,
except
h a v e at m o s t
we consider
in o u r
description
with
Q, t h a t
of
if
as
of
respectively.
number
that
section.
Nevertheless,
are dealt
same
the m e t h o d
transformation
number
Note
f l o w o n the
the c o r r e c t
countable
or a r c s
study of many
study of the
curve,
[Bir].
most countable of r a m i f i c a t i o n
the
to t h e
closed
complicated,
and of the corresponding
It f o l l o w s
most
induced
is a c o n v e x
be m u c h m o r e
known
be r e d u c e d
of the t r a n s f o r m a t i o n
In the c a s e w h e n
case
It is w e l l
of a f l o w c a n
only
of a r c s
~
piecewise {Fi}ia I
is a u n i o n
Ck
a n d of c l o s e d
the
representation
and curves,
curves F =
where
all
Ck
and all of a of
(i~lLi) Li
U
and
C k. F
has a fixed orientation
is fixed.
Let us
if t h e o r i e n t a t i o n
fix an o r i e n t a t i o n
of
F.
Let
of z
to some
L. or to s o m e F . Consider first the case when i 3 As L. is a n o r i e n t e d arc, o n e c a n c o n s i d e r t h e o r i e n t e d I ~z . Notice that when z is an e n d to L I at z, d e n o t e d h e r e
L i, o n e c o n s i d e r s
is o f t h e
first kind
kind.
In the
latter
to the
interval
the one-sided and
let
case we will
[0,2~).
tangent.
Let
e 6 S 1 = ~/2~
We denote
consider by
0 S 8 S ~
if
Li
if
9
(z,e)Li
L. is of the s e c o n d 1 as a n u m b e r b e l o n g i n g the unit
tangent
vector
205
at
z
which
has an oriented
angle
0
with
the o r i e n t e d
straight
line
£ . In a s i m i l a r m a n n e r , for z E F and 0 Z 0 < 2~, w e d e f i n e t h e z 3 v e c t o r (z,e)F . F r o m n o w o n w e w i l l a l w a y s s u p p o s e t h a t t h e o r i e n t a 3 t i o n of £ s a t i s f i e s the f o l l o w i n g c o n d i t i o n : the v e c t o r s (z,e)Li and
(z,@)Fj
uniquely kind
defined
defines
above
are directed
the orientation
and all closed
curves
of a l l
F
inside closed
(see Sac.
Q.
This
arcs
2).
If
L. 1 L
3
kind,
then
vector
for a l l
(z,0)Li
f r o m n o w on w e w i l l
omit
lead
to a n y a m b i g u i t y .
used
in
[Sin] 2
tangent
also
Define depending tively.
MFj
and
M =
two different the
= £j
MF~)
L. I
are metric
second
instead
and
F 3 with
angle of
the c o r r e s p o n d i n g the n o t a t i o n s ,
but
this will
oriented
the a n g l e
with
never
normal
is
an o r i e n t e d
[Bit]. x
[0,~] L• l
MFj
and
is of the can
= L1• x
MLi
first
be n a t u r a l l y
( U ML•) U (•~IMFj) ; q if ial z eiements o~-this summs
same vector
z 6 L 1 To s i m p l i f y
~.
that the
[BUn]l_ 4
in
for a l l
indices
Notice
on w h e t h e r MLi
Define
and
used
and
inside
the
first
is o f t h e 1
0 S 0 < 27
is d i r e c t e d
condition of t h e
TI~2,
then we
spaces
with
where
d(z,v)
[0,7]
or
or of the
second
considered
as
p =
(z,9)
q =
and
if
identify
them
the metric
and
p
p
and in
given
M L i = L.1 × S 1 respec-
o f T1 ~ 2
(v,~)
q
M.
kind
subset
belong
correspond All
by t h e
MLi
. to
to
(rasp.
formula
P(p,q)=
I
/[d(z,v)]2 on
Li
+ ~2
(rasp.
rectly
Fj)
and where
to t h e m e t r i c
considering
the p
p(p,q)
for a n y
component We
of M.
Notice
that
on the
the
is c o n n e c t e d
space
M
depends
representation
of an a t m o s t
countable
number
sional two
compact
(cf.
Sac.
Clearly, any
along
us n o t e
that
with
is a g l o b a l
smooth
to s o m e
boundary,
MLi
of
and
leads of
always
returns
3.3.
L e t us d e s c r i b e {Tt},
to
di-
M,
by
such a component. M
if o n e d e f i n e s
to t h e F
same connected
is c o n n e c t e d .
on t h e
region
a boundary manifolds and
finite
MFj,
number
then capacity
of
b u t a set
with i,
boundary j a i.
o f two d i m e n M
is e q u a l
for the b i l l i a r d
flow
in
in
and,
after
Q
meets
M
~,
i.e. finite
M°
now the i.e.
flow
the
v
( U~I F 4 J ). wi~h
compact
is a u n i o n
section
of the b i l l i a r d
time,
flow
M
U
z
l.ii). M
trajectory
billiard
of
(iUILi)~
This
of
iff
not only
a manifold
belonging
when
manifolds
F =
is n o t
arcs
between
component
space
belonging
M
M
together
and not
that
speaking,
Let
two p o i n t s
to the w h o l e
6 M
Generally
glued
on any connected
between
c a n be e x t e n d e d
stress
but also
path
p, q
the d i s t a n c e
~ = min(Io-~I,2z-10-yl).
defined
shortest
The metric = 1
p
denotes
transformation
induced
on
M
transformation
defined
by the
by t h e time
of
206
the
first
return
w =
(z,~)
6 M
line By
beginning zI
to
straight [z,z I] not
Let of
F.
= (Zl,@l).
same
6 M
trajectory
all
flow
denote
{Tt}.
the h a l f
to the v e c t o r
the n e a r e s t such that
contained
and
point
z ~ zI
in
S
TSw =
¢
with
with
by
z I = Zl(W)
and
that
trajectory
contains
parallel
L(z,e )
s = d ( z , z I)
L e t us d e n o t e S
and denote
of t h e
we will
~.
w =
If straight
(z,e).
of
F
and
such that
Note
that
belonging
to
the
such a point
a t all.
This means
a billiard the
line
(z,0)
Let
z
is e n t i r e l y
exist
w =
trajectories
L w = L (z,@)
we w i l l
interval may
along
at point
= Zl(Z,8)
the h a l f
zI
M
t h e n by
F
of d i f f e r e n t i a b i l i t y
We d e f i n e
~(z,~) =
to the c o l l i s i o n the
(see Fig.
the
the p o i n t s
associates
boundary P
be a p o i n t
(Zl,@l).
subsequent
(z,%)
of
collision
of
3).
s e t of e n d s of a l l
arcs
of n o n d i f f e r e n t i a b i l i t y
{ L i } i a I.
of
Clearly,
F.
Fl
Z
Fig. L1
3.
~(z,0)
and
We will
F1
belonging F=
( U L i) ial
(Zl,81).
at p o i n t s
consider
the point
=
z ~ S to U
S.
z
~(z,0)
Zz and
and zI
£z
are the 1 respectively.
not defined
or t h e p o i n t So,
the domain
( U F.). j~l 3
Generally
for a l l
z I = Zl(W) of
~
(z,@)
is e i t h e r
depends
speaking,
w =
oriented
o n the
the d o m a i n
not
tangents
E M
to
for w h i c h
defined
or
representation D~
of
¢
and
is
207
the d o m a i n liard
D~0
flow
{T t}
call
~
4).
De0
~
are not
.
Thus,
the mapping
It is e a s y
is a c o n v e x
transformation
closed
~
¢0
exactly
to see t h a t
same,
on
M
on
M
by t h e b i l -
but
it is f o r m a l l y
induced
: D ÷ M
induced the
although
a transformation
In g e n e r a l , Fig.
in
= ~0ID~ N
¢ID¢ N De0 we will
of t h e
by t h e
not correct, flow
is n o t a c o n t i n u o u s
the m a p p i n g
~
{Tt}. one
is c o n t i n u o u s
(see iff
F
curve.
L1 v
Fig. 4. The points t i n u i t y of m a p p i n g
Denote and
such
that
: M1 + M
4. 4.1.
by
M1
the
subset
e I @ 0,7.
is a
THE M A P P I N G
(z,e), (v,~) ¢ : D% + M.
THE
F r o m n o w o n we w i l l
next
two
with
the computation
subsections
and
of all
Clearly,
homeomorphic
%.
L~
(w,~)
(z,e)
M1
are points
E D~
is an o p e n
such
of d i s c o n -
that
subset
e ~ 0,z
of
M
and
imbedding.
COMPUTATION study
w
OF
d%.
the mapping
we deal with
9.
In t h i s a n d
the d i f f e r e n t i a b i l i t y
of t h e d e r i v a t i v e
d%.
This will
of
in the ~
l e a d us
and to the
208
well known
~
invariant
G.D.
Birkhoff.
Sec.
8 of
THEOREM
absolutely
[Bit] or be e a s i l y d e d u c e d
4.1.
Let
is of c l a s s
(P0,e0)
6 M1
Let
4, a < ~ < b
~i'
¢(P0,e0)
r
of c l a s s
be a p a r a m e t e r defined
Denote
by
(F(~),G(~)),
(FI(41),GI(41)). metric
(F(40),G(40)) Let
P of
(resp.
We will x-axis
an d
p0
angle to
of
pl = P1 )
(resp.
denote
by
between P1
r
T
the x - a x i s 5).
by
in Chap.
(Pi,01).
of points C k-I
VI,
p0
r
and
Ck
pl,
corresponding
of p o i n t
in a n e i g h b o r h o o d
F, G 6 ck(a,b)
P.
Let
of p o i n t
pl.
the c o r r e s p o n d i n g
in the n e i g h b o r h o o d
F 1 G 1 6 ck(c,d)
If
in some n e i g h b o r h o o d
of class
p0
and by
the c o r r e s p o n d i n g
in the n e i g h b o r h o o d
of
pl.
Let
para-
p0 =
(FI(~I),GI(41)). denote
% = e - T
(see Fig.
r
pl).
and the o r i e n t e d
to see that
of
41 6 (c,d),
representation
borhood
~ 6 (a,b),
representation
=
in a n e i g h b o r h o o d
c < ~i < d, be a s i m i l a r p a r a m e t e r
parametric
P
of
discovered
f r o m it.
c k, k ~ 2, in some n e i g h b o r h o o d s
to the o r i e n t a t i o n
measure
f ound e x p l i c i t e l y
and let
then ¢ is a local d i f f e o m o r p h i s m of (p0,~0).
Proof.
continuous
All this can be e i t h e r
a point
(resp. tangent and
f r om a s u f f i c i e n t l y
We s u p p o s e
t hese n e i g h b o r h o o d s
~i ) the o r i e n t e d to
F
in
eI = T1 - e
P
disjoint.
angle between
(resp.
where
and the t r a j e c t o r y
small n e i g h -
~
Pl ) .
the
It is easy
is an o r i e n t e d
of the b i l l i a r d
going
from
209
T1
£PI
Fig. 5. @ = ~ - T, 01 = T 1 of F at the n e i g h b o r h o o d s is of no importance for the d i f f e r e n t i a b i l i t y of F in is required.
Rotating, suppose that
if necessary, F' (~0) ~ 0
-~. On this figure the b o u n d a r y pieces of points P and P1 are convex. This proof of T h e o r e m 4.1 where only the C2 some n e i g h b o r h o o d s of points P and P]
the region
and that
g e n e r a l i t y it can be supposed that s u f f i c i e n t l y small n e i g h b o r h o o d s of It is also easy to see that G i (~i) Fi (~i) .
Then :
~
on the plane, one can always
Fi(~')
~ 0.
F' (~) ~ 0 ~0
and
G' (~) tan T = F - ~
W i t h o u t any loss of and
#i
F~(#I)
~ 0
in
respectively. and that
tan T 1 =
Thus, t a k i n g i n t o a c c o u n t t h a t
-
8 = a
T and
el
G(m)
def
-
-
T~
-
a , we
o b t a i n from ( 4 . 1 . ) :
G; =
Arctan
From ( 4 . 2 . ) equations f o r
(ml)
-F; (Q1)
and (4.3.)
$1 = $ l ( Q , 8 )
G1(Q1)
Arc t a n
F1(Q1)
-
F($)
M(m,ml)
(4.3.)
we o b t a i n t h e f o l l o w i n g i m p l i c i t e f u n c t i o n and
el
= e1(@,8):
A c c o r d i n g t o t h e I m p l i c i t F u n c t i o n Theorem, f o r t h e . e x i s t e n c e o f Ck-l = $ 1 ( $ , and e l = 81($,8) of c l a s s satisfying -
functions
t h e equations (4.4.)
-
(4.6.)
i n some n e i g h b o r h o o d o f
sufficient that:
where but
"$1 -
aA ,
Ael=- aA, etc.,
A&1 = Lelr Ael = 0 , Bbl = M+l Thus, we o b t a i n t h a t :
and
Bel = -1.
($Or8O)
it i s
211
Let us compute respect
to
¢i
L~I(¢0,80).
By differentiation
F{ (~l)
G~ ( ~ I ) )
F(~0)-FI(¢I) L¢I(~0,80
(4.8.)
det
L~I(~0,80)
{ kF(~0)-FI(~I)
This is equivalent
pl
to
¢
what has been proved
pl.
of
This
above, C k-I
arcs
{L i}
of
and because
F
of class
and all closed curves
: M1 ÷ M 4.2.
4.1.
obtains
If the boundary
is a C k-I
F
diffeomorphic
It is quite natural
obtains ¢-i
from
is also
(pl,@l).
This means
in some neighborhood •
C k, 2 ~ k S ~, then all
{rj}
are also
the following
C k.
From
is piecewise
in the neighborhoods
C k, 2 s k S ~, then
imbedding.
suppose
of points
p0
and
pl
on
length of arcs
from some fixed points
F
measured
formula
that the parameters
fixed arc length parameters Let us proceed
corollary.
then to find an explicit
From now on we will always defined
that
in some
j~l
4.1. one immediately
COROLLARY
and
one considers
that
of C k-I
be a piecewise
i~l Theorem
(p0,80),
G 0 # 0,z,
P
we assume C k-I
then one immediately
in some neighborhood
that ¢ is a local diffeomorphism of (P°,80). Let now the boundary
of class
and
one obtains
through
is so because
is a mapping
(Pl,81),
Finally,
line passing
in a neighborhood
in a neighborhood of class
This
(F{(¢I),G{(¢I))
are not parallel.
~
of
zero.
0
iff the straight at
a mapping
is never
to the fact that the vectors
F
If, instead
(4.8.)
GiIl) 1
(P0,00) 6 M I. Therefore, neighborhood of (P0,80). ¢-i
of
, G(~0)-GI(~I) /
L~I(~0,80 ) ~ 0 is not tangent
[G(#0)-GI(¢I)] 2
~ 0 iff
(F(~0)-FI(~I),G($0)-GI(¢I)) that
+
p ~ p1, the denominator
that
, G(~0)-GI(# I)
= [F(¢0)-Fl(#l)]2
Since
(4.2.) with
one easily obtains: det (
implies
of
formula
for
~
d#. and
respectively
F, i.e. the parameters
now to the Benettin
for
of
~i are
given by the F.
de(P,@)
which
the basis of all future considerations. From now on we will speak very often without
any distinction
of
is
212 Points
P
dering
d~(P,8)
and
PI'
and
Let us i n t r o d u c e k = k(4),
k I = k(4 I)
at p o i n t
z.
points
P
THEOREM 91
By
and
as b e f o r e
9~ and
91
we w i l l c o n s i d e r the f o l l o w i n g where,
i.e.
k(z)
Instead
of c o n s i -
etc.
notations:
recall,
i = Z(P,8) Pl'
respectively. d ~ ( 4 , 8),
d = sin 8, d I = sin @i'
denote
we w i l l d e n o t e
the c u r v a t u r e
the d i s t a n c e
~ = /[F(4)-F(91)] z +
of
between
[G(4)-G(41)] z
F the
where,
¢(p,o)=(Pl,el).
4.2.
Suppose
respectively
¢(90,80 ) =
F
(41,8 l) .
small n e i q h b o r h o o d
t h a t in som____~en e i q h b o r h o o d s is a
C2
curve.
Then,
for
Let
(9,8)
o_~f (40,80 )
of p o i n t s
(90,80 ) ~ M 1
belonqinq
40
and
and let
to a s u f f i c i e n t l y
one has:
~(9,e) , ~(9,e)
d~(9,@)
~e I
(4.9.)
~e 1
-~--$--(¢,e)
, -f~-(¢,e) %
kS - ed el dl ed
-
elklk~ where
Z ' el dl kl£ --k , e I dl
dl
parametrised
e I = el(90,@ 0) are the c o n s t a n t s In a p a r t i c u l a r
arc or s m o o t h l y
We p r e c e d e
4.1.
o_~f r.
(4.10.)
1
e = e(90,@ 0) = ±i and
on the o r i e n t a t i o n
LEMMA
)
parametrised
the p r o o f of T h e o r e m
Under
the c o n d i t i o n s
4.2.
case w h e n
F
c l o s e d curve,
4.2.
e = e I = i.
lemma:
one has:
L¢
=
d sT - k
(4.11.)
L¢I
=
d1 el-~-
(4.12.)
M9
=
d -s~
(4.13.)
Me i
=
d1 kI - eIT
(4.14)
where
e = ±i
and
the o r i e n t a t i o n metrised Proof.
of
E 1 = ±i F.
arc or s m o o t h l y To p r o v e
are a b s o l u t e
In p a r t i c u l a r parametrised
the f o r m u l a s
(4.11.)
constants
case when
F
c l o s e d curve, -
(4.14.)
only
i__{ss m o o t h l y then
by the f o l l o w i n g
of T h e o r e m
dependinq
dependinq
o n l y o__nn
i__sss m o o t h l y then
para-
e = e I = i.
one uses the c o n s e c u t i v e
213
differentiation
of formulas
Let us prove for example
(4.2.)
- (4.3.)
the formula
and the formula
(4.12.).
The other
(2.1.).
formulas
are
proved analogously. We will use the formula
(4.3.) which gives the value of
in the form of a fraction. to Z2.
The numerator
The denominator
is equal
of this formula
to the oriented
gram spanned by vectors (F~(¢I),G~(¢I)) and The length of the first vector is equal to 1 length parameter Thus,
on
one obtains
r.
, G' (¢i)
F(¢)-FI(¢ I)
, G(¢)-GI(¢ I)
is a constant
that
Proof of Theorem a) 0
Using
(4.15.)
depending d = c i ~.
L¢I(¢,@) 4.2.
Calculation
Let us calculate
of
-~.
I.
L¢I-~- = 1
one finally
(4.15.
ClZ sin 01 = el£d 1
only on the orientation
of •
3¢1 3e 3e ' 3¢'
Differentiating
3¢1
one obtains:
to
) =
gl = ±i
(F(¢)-FI(#I),G(¢)-GI(¢I)). because ¢ is an arc
that:
F' (¢i)
This implies
is equal
area of the parallelo-
The length of the other one is equal
det
where
L¢I(¢,0)
and by consequence
3¢1 3¢
(4.2.)
3¢1 38
361 3e "
and
with respect
to
1
- L¢I(¢,¢ I)
obtains:
3¢ 1 3e (¢,e) = £ i ~ b) respect
Calculation to
361 _ 3¢
(4.16.
¢
of
381 3¢ "
and using Lemma
1
det/ L¢I
L¢I
I L¢
Differentiating 4.1.,
(4.4.)
/ L¢I
' Me1) =
dl)
Me
=
L¢
=
glk I (ki - ed
91
c) ¢
Cl~ll det ( ~ id-~-
=
k~[ - k , - gd
, Me
)- k.
Calculation
one obtains
' kl - £i-~-
' Me1
t
(4.5.) with
one easily obtains:
d1 Z = gl~ll det
and
that
of
3¢ 1 3¢"
Differentiating
(4.3.) with respect
3¢ 1 L¢ + L¢l ~ = 0, and by consequence
3¢1 3¢
to L¢ L¢ 1
214
Thus
from
(4.11.)
and
(4.12
) one o b t a i n s :
,
d) O
~($i~)
= elk£ - Ed dl
•
Calculation
one o b t a i n s
~91 ~.
of
that
~6
Differentiating
- M$1
~O "
Thus,
(4.4.)
from
with respect
(4.14.)
and
to
(4.16.)
one
obtains: ~91 5-6 (¢1'6)
4.3.
=
i.
Let us fix o n c e and
for e x a m p l e Theorem
pieces
of
•
for all a n o r m
(a
,
b)
c
,
d
the n o r m
4.2.
COROLLARY
4.2.
Let
F.
P ( I; e < 6( (P,8),
=
we i m m e d i a t e l y
Let
o f the ends of
near
klZ di
s
I
and
P
J.
obtain
J
Let
~
following
and w h e r e
•
= ~i.
Then :
the b o u n d a r y
at
closed
Let (P,@)
boundary
~i
(P,9)
From
consequences.
C2
a n d let
be one
( M 1 where
is s u f f i c i e n t l y
Let lim IId}(¢,e) JI = + (P,e) ÷ (P,O) P6I, (P,6) (M 1
~i.
9 > 6.
through
A completely (See Figs.
--
P
--1
an_~d P
analogous
6a - 6e).
J
Fig. 6a
Idl •
matrice,
e sin
81
sin
81
< -
fp(X) x x2
Mean
xf~(x)
- fp(X)
Value
Theorem
2 x xf~(x) - fp(X)
eI 81
P
x + fp(X)
tan
has
between
that
81
the C a u c h y
one
the quan--
tan ~ - tan @ 1 + tan ~ t a ~ - 8
!
tan
8 > 0
is the d i s t a n c e
e I = tan(~-8)=
fp (x) x
fp(X)
tan
sufficiently
as we recall,
!
8 =
for
fp(X) 1
sufficiently
0.
Let us e s t i m a t e
sin
this
l i r a .(i) (x) = g(i)(x) P~P rp
de f h. L e t us s u p p o s e
tan
that
1 fp(X)"
As
Qne
obtains
2 = f!'p(~)" tan l i m sin 8+0
81 @ 1
1
and
8>0 of
formula
(4.10.)
for
de,
one
P(Y+ deduces lim
IId#(P,8)ll
Let n o w Then,
as
y
tan
0 -< lim
> £(P,8) - sin 81
y
is of c u r v a t u r e
1 l i m . . =
~p~x~
Z(F,O)
N o w we w i l l
< ~
O 1 - g" (0)
prove
that
< + ~.
at
i.e.
let
P,
then
+ ~.
be of non zero c u r v a t u r e 1 01 > - ~ ( t a n {3 - tan 8) one
sin
zero
at
P,
g" (0) ~ 0.
obtains
(4.17.)
218
lim
sin
sin 8 01(8,P)
=
tan l i m tan 8 8÷0 s-~n ~ - lim sin
As
(4.18.)
i.
81 (PI,9) 81 (p,9)
= l, to d e m o n s t r a t e
(4.18.)
it is
9>0 sufficient
lim
to p r o v e
tan
that
tan 8 81(P,8)
=
(4.19.)
i.
fp(X) But
tg 8 -
and
X
tan
fp(X) x
f~x
From
fp(X) i + f~(x)---~
this,
fp(X) =
as before,
xf~(x)
=
- fp(X)
fp(x)
lim
tan e tan 81(P,9)
quently
= 0
and
Now,
lim
-
X
-
f'(x) = lim P _ i _ _ x
(4.18.),
(4.17.),
has
tan
tan 8 81(P,8)
fp(X) f~(~) •- x ) "~ "
(l+fp(X)
are
proved.
(4.18.)
let us
and
= i.
N o w our
assertion
formula
another
4.3.
If in the a s s u m p t i o n
small
neighborhood
to
U
preserves
measurable
Proof.
The
subset
U
(4.10.)
of
~0(S)
du O =
one
But
has
(4.19.),
and c o n s e -
immediately for
follows
d~ .
of T h e o r e m
Theorem
of the p o i n t
the m e a s u r e S c U,
Then
consequence
ficiently stricted
this,
1 lim f ~
from
formulate
From
g"(O).
COROLLARY
every
one
f~(x)
!
lim
4.4.
just
fp (x)
( l + f S ( x ) ' ~ )
from
tan 8 - tan 9 = 1 + tan ~ tan 8 =
81 = tan(6-@)
4.2.
we c o n s i d e r
(P0,80), Isin
4.2.
then
81dSd%,
%
i.e.
for
f o r m u l a (4.10.) for d% immediately implies that sin 8 . ~ Now, the C o r o l l a r y 4.3. f o l l o w s f r o m
of c h a n g e
From always end of P0(M)
of v a r i a b l e
the d e f i n i t i o n ~0(M\MI)
Sec.
3.3.
< + ~ iff
COROLLARY
4.4.
in d o u b l e
of m e a s u r e
= 0, w h e r e We
recall
IFI
< + ~.
If
IFI
M1 that
< + %
~
then
the
integral. one
is the IFI
re-
= ~0(¢(S)).
Idet d%(¢,@) I = formula
a suf-
obtains
subset
denotes
of the
the p r o b a b i l i t y
immediately M
defined
length
of
measure
that at the
F.
Clearly
219
d~
-
1
Isin 8 I d e d ¢
~0(M)
S c M
subset
one
Corollaries Sec.
8 of
det
formulas
imply
(4.1.)
Proof.
invariant,
go b a c k
Birkhoff
to G.D.
proved
of the m a t r i x using
every measurable
the
Birkhoff
Corollary d~(¢,8),
(see Chap.
4.3.
without
VI,
explicit
but he computes
formulas
obtained
by d i f f e r e n t i a t i o n
4.4.
the B i r k h o f f
of
(4.3.). that
that
Corollary
m(Z)
If
= 0, w h e r e
IFI
for w h i c h
< + ~,
the c a s e s
respectively. that
For
for
= U(~(S)).
4.4.
and
~
then
Z = Z A U Z B U Z C, w h e r e
occure
i.e.
denotes
the
Ergodic
Lebesgue
Theorem
measure
× S I.
4.5.
v ( TI~
prove
-
note
easily X : ~
COROLLARY
G.D.
directly,
Finally,
~
u(S)
and
of e l e m e n t s
d¢(¢,~)
in
has
4.3.
[Bir]).
computation
is
Y(Zc)
this
~(Z)
Z A,
A,
Clearly,
B
=
ZB
and
and
~(Z A)
0.
C
ZC
are
sets
described
= ~(Z B) = 0.
of
such
in Sec.
3.1.
It r e m a i n s
only
to
= 0.
purpose
it is e n o u g h
to p r o v e
that
U ( Z c N M I) = 0, as
U (M I) = i. For
w =
of e a s e
C
~(~kv)
(z,0)
6 M
we will
it f o l l o w s < + ~.
that
note
for
v
In p a r t i c u l a r ,
~(z,8)
E M1
for
= ~(w).
one
has
By definition
v ( ZC
v E Z C N MI,
one
iff
has
k=l 1 n lim ~ ~ ~(¢kv) n÷~ k:l that
= 0.
~(ZcNMI)
to the
space
to the
%
to the
= c > 0. ZC N MI,
invariant
function
everywhere,
Clearly, Then, to t h e
/
M1,
I J
~du c > 0.
invariant
= I
Ergodie
~(~kv)
Z*du c.
set.
Suppose
Theorem
applied
% : Z c N M 1 ÷ ZC A MI,
1 ~c = c U
measure
lim n [ n÷~ k=l
ZcNM 1 on
~
transformation
gets
ZdUc
is a
from the Birkhoff
probability
i, o n e
and
ZC N M I
=
As
on
Z C N MI,
and
(v) , U c - a l m o s t
£
is s t r i c t l y
positive
ZcNM 1 This
is in c o n t r a d i c t i o n
with
the
fact
that
ZcNM 1 ~*(v)
= 0
Thus, where
on
measure
on if M ~.
Z C N M I. IF1 and
< + ~, ~
In t h e
So,
U ( Z c A M I) = 0.
the mapping
preserves future,
¢
•
is d e f i n e d
the a b s o l u t e l y
when
speaking
u-almost
continuous
about
ergodic
every-
probability properties
of
220
~, we w i l l
4.5.
consider
Let us m a k e
{F i}
and
n o w a few r e m a r k s
{Lj}
ial (4.10.) If H3,
them exclusively
with
about
, a n d of the n u m b e r s
respect
to the m e a s u r e
the o r i e n t a t i o n e
and
eI
u.
of the c u r v e s
(see
formula
jal associated F
... and
orienting
them. of m u t u a l l y {Hi ~ 22
if all c u r v e s
all
boundary
with
is the u n i o n
the c u r v e s
pieces)
one o b t a i n s
disjoined are
inside
closed
curves
the c u r v e
{H i }
(and t h e r e f o r e a l s o i~2 in the d i r e c t i o n o p p o s i t e to t h a t of
HI,
H 2,
H I, then their H1
smooth
(see Fig.
8),
e = e I = i.
H3
Fig. 8
Nevertheless,
generally
a
such
that
eI
are
(z,8)
ively,
6 M e
precisely
and such
speaking, in some
in o t h e r
neighborhoods
of o p p o s i t e
a situation).
cases
signs.
of
one z
(See Fig.
could and
always zI
9 which
find
respectshows
221
z ¸
Fig. Notice ments
also that
if one changes
of the m a t r i x
approp r i a t e l y . complete
on
elk I r
4.6.
5.1.
the m a t r i x
of
r, then
(4.9.)
change
with
their
of signs
is in
the rule of t r a n s f o r m a t i o n
the o r i e n t a t i o n
do not depend
of
r.
the o r i e n t a t i o n
Indeed, of
r
of m a t r i x
the elesigns
(4.10.)
the q u a n t i t i e s
but i n t r i n s i c l y
ek depend
only.
Finally,
defined which
the o r i e n t a t i o n
i.e.
It is easy to see that this c h a n g e
agreement
when one changes and
d¢(~,0),
9
let us note that very
in this
section
the results remain
true
large classes
do not yet e x h a u s t
described (see Fig.
in this i0).
section
of plane
the class as well
regions
of regions
for
as those of Sec.
222
-
Fig.
i0.
F o r all
k,
H
is t h e o n e p o i n t
5.
APPLICABILITY
5.1.
From
sidered
Appendix then
n o w on w e w i l l have
we w o u l d
first
to t h e m a p ¢.
C2
in Secs.
5.2.
billiard
whose
of c l a s s
Ca
boundary. - 5.4.
of
Since
Pesin
However,
not
theory
always
satisfy
an e x a m p l e C1
it,
THEOREM that
all con-
to the b i l l i a r d s ,
the O s e l e d e c
such a class
describe
rectangles.
length.
in the O s e l e d e c
is of c l a s s
does
of
almost
ERGODIC
repeating
finite
the
disjoint
U Fk k~l
without
consider
we w i l l
boundary which
assume,
to a p p l y
of
the a p p l i c a b i l i t y
f r o m n o w o n we w i l l
piecewise
are mutually
MULTIPLICATIVE
boundaries
like
to d i s c u s s
2)
Hk
compactification
OF T H E O S E L E D E C
billiards
Since have
1 ~ k < ~,
Tl
and
theorem
theorem only
one
uses
billiards
is too
large
the a s s u m p t i o n
for o n e of
d~,
with
because
of a s i m p l y
except
we
(see
connected point
also
the Oseledec
theorem.
THEOREM
5.1.
a r y of f i n i t e boundary
us c o n s i d e r
length.
is u n i f o r m l y
to t h e m a p p i n g
Proof.
Let
We are
If the a b s o l u t e bounded,
and
that
then
with
value
a piecewise
C2
of t h e c u r v a t u r e
the O s e l e d e c
theorem
boundof
its
is a p p l i c a b l e
¢.
to p r o v e
I log+]jd~(¢,0)ljjsin M
a billiard
that
01d~de
< +
(5.1.)
223
fMlOg+H[d%(¢,8)]-iNlsin
where
log+a = max
Denote
by
h(%,8)
=
h
81d%d0
(5.2.)
< +
(0, log a).
the m a p p i n g
of
M
onto
itself
given
by the formula
/
Clearly,
~(%,~-0)
for
0 ~ 8 ~
< (~,3~-8)
for
z ~ 8 S 2z
h = h -I
perty").
and
Moreover,
ho~o h = ~-i
as
dh(~,8)
=
(o)
(the so c a l l e d
0 all
(%,8)
6 M.
From
the time
lldh(~,8)II : 2, it follows (5.2.).
Therefore,
F r o m the has
formula
IId¢(%,8)I;
4 1 5
x
Fig. 12 Let us d e f i n e fa the
6 C~([-a,a]) same
for and
structure
0 < a < 1 in the
as the
fa (x) = a 3/2
interval
function
[a-,a] f
on the
f(~). the
Clearly,
function
interval
fa
[-i,i].
has The
226
only
difference
tion
is an arc From
is t h a t on the of an e l l i p s e
the u s u a l
h"(x) (l+[h, (x)]2)3/2, absolute less
that The
Fig.
values 1 7~"
will
interval
I =
For
interval U 13 U J3
of
that
of
the g r a p h
of
func-
t h a t of a c i r c l e . of c u r v e
for all
the g r a p h
3 3 (-~a,~a),
x E
of
y = h(x);
function
k(x)
piece
L
laying
between
be c o n s t r u c t e d
in the
[0,4d]
d =
I
where
by
can
In
and
~ n=2 Jn
be r e p r e s e n t e d
J~
13
the points
following
U I n U Jn U ....
I
the
f
is n o t
r
and where
n
!
n
-
intervals
as a u n i o n
of
as
shown
..........
B
(see
1 n ( l o g n) Z" of
intervals
on Fig.
In
!
r
and
L e t us t a k e a n
the closed
J3
!
A
manner.
length of
2r n.
12 U
13.
Jn
I
"
¢ ......
Fig. 13
Define
o n the
interval
I
the
function
g
as
follows:
n-i
~ rk-rn) Ii 3n/2frn k=l (x-4
g(x)=
where,
by d e f i n i t i o n ,
function
Fig.
2
x E In,
n { 2
for
x E Jn'
n { 2
the c u r v e
The
shape
It is e a s y not have
the this
the c u r v e
L
. . . . . . . . . . . .
I3
J2
14.
Indeed,
Define
r I = 0.
for
L
as a g r a p h
of
g.
.3~k 1
=
a
U ...
12 L
easily
of t h e c u r v a t u r e
n { 2, d e n o t e
The J2
instead
for the c u r v a t u r e
it f o l l o w s
boundary
ii)
formula
3 3 [-~a,~a],
interval
J3
of
to c h e c k
second follows
the g r a p h
that
of
immediately
in the c e n t e r s
of
: .......
In Jn
function
g E cl(I)
left derivative
:/h:
g
but
g
~ C2(I),
at the r i g h t
from
the
intervals
fact In
e n d of
that tends
as
g
does
interval
I.
the c u r v a t u r e
of
to i n f i n i t y
when
227
n
tends
to t h e
Note
5.4.
To
of o u r and
that,
infinity. since
finish
II
the c u r v e
the c o n s t r u c t i o n
rectangle,
to v e r i f y
g (CI(I)
the
size
of
of
F
L
is of f i n i t e
it r e m a i n s
smoothing
of
length.
to d e f i n e
the c o r n e r s ,
the
the
size
set
H,
that
(log k(~)
sined~d~
= + ~
(5.5.)
H
L e t us m a k e
following
class
C1
on
one can consider
L
of
the
F
which
to
see t h a t
finite
associates
length,
remark. instead
Since of a n a r c
an e q u i v a l e n t
a number
x
L
is a c u r v e length
parameter
parametrization
to t h e
point
of the {4}
of c l a s s
(x,g(x)).
C1
on
It is e a s y
fr
where
and
by
k(x)
is e q u i v a l e n t
we denote
L e t us d e n o t e
by
that
e n d of
the
val
In .
that
the
point
left
Consider ray
there
x 6 Kn,
Kn
an K
between exists
Denote
one has
A =
7 > 0 6(x)
(log k(x))
such
by
of
L
with [
3 ~r n
= +
of
15). L.
inter-
vector L
We denote
n ~ 2
K n ¢ In
the
unit
to t h e c u r v e
and vector for a l l
(x,g(x)).
such that
the c e n t e r
the u n i q u e
(see Fig.
that
0dx dO
at point
length
is t a n g e n t
(0,i)
sin
H
of
coincides
y 6 In+ 1
vector
~ JJ
interval
n
x E K n.
where
to
the c u r v a t u r e
{ (x,g(x))+tk,t~0}
(y,g(y))
the a n g l e that
(5.5.)
by
6(x)
It is e a s y and
such
at a
to see
for a l l
~ y.
A
~
(x) >
!
x
,
k,
J
0
probability
described
Let
P3
of c l a s s
torus This
billiards.
theorem.
C 2.
non-zero
and
then and
it is n e c e s s a r y
{L i}
billiards
in a n a t u r a l
P2'
of the m a p p i n g .
to b i l l i a r d s ,
curves
will
following
C1 > 0
invariant
iEiEp 7-8 this
the
has
OF C L A S S
7.1.
with
directly
4.4.
THE B I L L I A R D S
liards
imply
is a b i l l i a r d M
0 < s < s 0, one
in the C o r o l l a r y
7.
(6.1.)
~ of
(6.1.
the b o u n d a r y from Corollary is e q u i v a l e n t
contains 2),
a point
of
that
manifold.
a strictly
or a s t r i c t l y zero c u r v a t u r e ,
238
or a s t r i c t l y
convex
Nevertheless, class and
H
which
which
are
near
the
fast
results This
from
enables
are
The
class tion
of
a particular
set.
I-IV
us
deduce
true
convex
of
curves
on w h a t
denote
by
s ~ t,
non-zero
f
the
for
f' (s)
and
Let on
> 0
of
E Via).
interval
every Rf(s
s, t)
Any
such
val
t E
f(s)
positive
symmetrical implies
that
for
[0,a] and
instead and
f
To Sec.
F
7.2
theless, can with
on a
in
replace only
results
class
of
of
E
~
all ~.
from
a particular
now.
This
F.
set
strictly
all
that
class
Pesin
uses
= f' (0)
too
as w e l l .
to d e f i n e
of
grow
immediately
billiards
condition
one
defini-
convex
=
0.
For
such
and
if
s,t
E
E
satisfies
the
a number
C > 0
+ f(t)
condition such
F that
> C
f' (t)
(7 i.)
-
be c a l l e d
s,
t
[-a,a],
has
(s-t)
will
f
exists
every
s
an
and
exponent t
conditon
[-a,a],
play F,
s ~ t,
one
the
of
f
(on i n t e r -
a completely inequality
(7.1.)
has
1 Z ~ < + ~
i7.2.)
0 < C ~ i. the
interval
and
the
such
intervals
only
strictly
[0,a] made
the
[-a,a]
that
f(0) or
the
Proposition interval
7.1., [-a,a]
modifications.
F
considers
convex
functions
=
f' (0)
= 0t o n e
on
the
leave
the
intervals
f E cl(i[0,a]) obtains
the
con-
respectively.
repetitions,
Theorem by
We
one
I-a,0]
to a v o i d
condition
the
minor
follows the
f E V(a)
that
of
framework
- 7.4
thus
definition
of
P
class
f(s) - f(t) _ f'(t) are both s - t (s-t)f' (s) - f(s) + f(t) > 0 f(s) - f(t) - (s-t)f' it)
in t h e
[-a,0]
fix
if
as
E cl([-a,0]),
dition
-
C
of
and
- f(s)
remark
of
IId2¢II c a n n o t
it
f(0)
that,
necessarily,
If
f' (s)
us
0 < C S Rf(s,t) Thus,
that
say
-f(t)
billiards
0.
s ~ t
number
Let
role
the
that
the
billiards
to
going
a > 0,
if t h e r e
[-~a],
de f (s-t)
[-a,a]).
are
kind.
for
of class
the
sign;
We will
the
of
call
clear
same
7-9
principal
billiards
x ~
[-a,a]
'
this
- f(s) - f(t) s - t
the
of
billiards
we
second
IId¢ll a n d
the
such
for
geometrically
then
the
V(a),
f(x)
It is
that
which
functions
case
7.8.
we will
f E cl([-a,a])
the
applicable
the
functions
of
in Secs.
From
are
for
definition
us
in Sec.
singular
to
piece
is p r o v e d
Parts
is b a s e d Let
it
defined
[PeS]l_ 3 remain
7.2.
boundry
as
the
7.1.
and
interval this
we will
interval
to
consider
[-a,a].
Corollary [0,a]
the
or
reader.
in
Never-
7.1.
one
[-a,0]
239
Let us n o t e
that
2 ~ i s k - i, a n d f(k) (0)
condition
F
of a f u n c t i o n is g i v e n
I f(x)
For
from
by a w e l l
V (a) known
e I/x2
for
0
0
condition
In the p r e s e n t
sibly
largest prove
class
the
7.1.
Then,
there
exists
tion
F
on the
7.1.
on any
we do not
interval
f (V(a)
a0,
Let
the
[-a,a].
such
function
Indeed
f 6 ck+2([-a,a])
Corollary.
COROLLARY
7.1.
interval
[-a,a],
If
f
satisfies
the pos-
F.
We w i l l
a > 0, and f
let
satisfies
be r e p l a c e d
f
a > 0, k { 2, Then,
there
the c o n d i t i o n
in T h e o r e m
f (V(a), then
for
that
f(k) (0) ~ 0.
compactness
following
such
the
Theorem.
N ck+2([-a,a]),
let
whether can
simple
and
search
the c o n d i t i o n
f"(0)
~ 0.
the c o n d i -
[-a0,a0].
f ~ V(a)
that
with
N C2([-a,a]),
0 < a0 ~ a
and
deal
satisfying
Proposition
interval
[-a0,a0]We do not k n o w
By the
paper
Let
2 E i Z k - 1
0 < a0 E a
F
of f u n c t i o n s
following
PROPOSITION
THEOREM
for
by M. M i s i u r e w i c z ,
= + ~.
7.3.
only
as r e m a r k e d
= X
for
N ck~-a,a],
=
this
Sec.
f 6 V(a)
f(k) (O) ~ 0, t h e n
> 0.
An e x a m p l e
more
if
if
7.1.
F
on the
a > 0
satisfies
one
and
if
f(i) (0) = 0 a 0, interval
the a s s u m p t i o n s
by the a s s u m p t i o n
argument
let
exists
obtains
f
f ~ ck([-a,a]). from
Theorem
is r e a l - a n a l y t i c
the c o n d i t i o n
F
7.1.
the
on the
on the w h o l e
240
interval
Proof
of
exists one
[-a,a].
Proposition a0,
f" (s) f"(t)
has Let
us
-f(s) + f(t) Value
G(t) H(t)
- G(S) - H(s)
As
f E C2([-a,a])
such
that
for
all
s E [ - a 0 , a 0]
and
let
us
and
s,
f"(0)
t E
and
H(t)
Theorem
= f(s)
one
has
G' (tl) H' (t I)
- f(t)
-
note
(s-t)
G(t)
=
to t h e
there
s ~
f' (t).
(s-t)
t
Then,
open
Rf(s,t)
f' (t I) - f' (s) tI - s
interval
linking
f' (s) + from
Cauchy
(s-t)f' (s) - f ( s ) + f(t) _ = f(s) - f(t) - (s-t)f' (t)
t
G(t) H(t)
_
1 • ~
for
some
f' (t I) ing
> 0,
[-ao,a0],
> ! - 2"
fix
Mean
7.1.
0 < a 0 S a,
and
s.
tI
belong-
- f' (s)
But
= tI - s
f " ( t 2) s.
for
Thus,
Rf(s,t) The
the
belonging
to t h e
open
of
Theorem
if o n l y 7.1.
s,
7.1.
If
k > 2
condition
F
Let
has
s,
Rf(s,t)
k
is e v e n ,
r ~ i. other
on
t E =
The hand
use
is b a s e d
is e v e n , the
then
interval
[-i,i],
on
the
the
continuous
~(r)
> 0.
and
we omit
The
the
and
7.1.
and
7.2.
for-
t ~
0
that
function
Rf(s,t)
> 0.
de ~Hospital
rule
function
= x
k
satisfies
[-i,i].
s ~ t,
= k - i.
f(x)
and
Thus,
¢
o n ~-- = ~
s r = ~.
For
f(x)
= x
(k-l)rk - krk-I + 1 def k r - kr + (k-l) Consequently, gives can
~(r)
> 0
l i m ~(r) = i. r÷l
On
be c o n s i d e r e d
U {-~}
U {+~}
k
~(r)
for the
as a s t r i c t l y
and
consequently I
following
7.2.
such
s ~ t, the
l i m #(r) r÷±~
inf rE
be
tI
•
Lemmas
function
(k-l)sk - ktsk-i + tk k - kst k-I + (k-l)t k
for of
positive
LEMMA
linking
t E [-a0,a0]
s As
interval
below.
Proof. one
t2
f"(t 2 ) ! > - f . ( t l ) - 2'
proof
mulated
LEMMA
some
finally,
lemma
is a s i m p l e
consequence
be e v e n
let
of
the
Taylor
Formula
its proof.
Let
k ~ 2
f(i) (0) = 0 g
given
by
for the
and
f E V(a)
2 S i ~ k - 1 formula
and
N ck+2([-a,a]), f(k) (0)
~ 0.
a > 0, Then,
241
is o f c l a s s The class
function
the
proof
As
v
of
(
the
b = g(t),
a,
In v i r t u e
h
some
q0'
~
inverse
=
As we
know
condition C1 > 0
F such
RF (a'b)
as
the
function basis
Lemma
7.2.
we
write
k[~(s)]k-lg' [g(t)] k -
bk
-
-
of
from
for
that
= ,
> O.
exists
> 0.
and
Moreover,
for every
us n o t e
a = g(s)
and
write:
(s)
-
[g(s)]k
+
[g(t)] k
k[g(t)]k-lg
' (t)
ak + bk
k b k - i [h (a) _ h (b) ] h , l (b) -
( a k - b k ) h ' (a)
def
- kb k-l[h(a)-h(b)]
. Tf(a,b)
Lemma
the
h = g
-1
=
h' 1(a)
( a k - b k ) h ' (b)
h' (b)
g,
can
g'(O)
[h(a)-h(b)~
kak-l[h(a)-h(b)]
h' (a)
where
Let
now
h' (a)
on
=
We
h' (b)
that
the
is t h e
so s m a l l < 2.
[-q0,q0 ] .
kak-l[h(a)-h(b)]
=
f
[ - q 0 ' q 0 ]' q 0
-< 1
h' (u) < 2 - h' (v)
[g(s)] k -
k
7.2.
idea
to
interval
0 < q0
[h(a)-h(b)]
a
in L e m m a
C k+l.
This
and
!
has b
of
(s-t)f' (s) - f(s) + f(t) f(s) - f(t) - ( s - t ) f ~ (t)
=
that
function
z = g(x).
E C2([-a,a])
on
one
=
Rf (s,t)
7.1.
g
We chose
where
the
> 0.
7.1.
C2
[-q0,q0 ]
by the
to c o n s i d e r
function
class
> 0.
class
where
> 0,
of
shows
replaced
Theorem
k
k/f (k) (0) k'
g' (0) =
variable
Theorem
[g(x)]
g ' (0)
h' (0) u,
of
Proof
is a l s o
-a _< x < 0
= x k + x k + l l Ix "
be
L e m m a 7.2. a l l o w s k = z in t h e n e w
=
for and
f(x)
cannot
F(z)
f(x)
0 _< x _< a
C2([-a,a])
C k+2
7.4.
for
7.1.
the
interval all
a,
function
[-i,i]. b
(k-l)a k - kba k-I bk (k-l) - kab k-I
6
This
[-i,i], + bk k + a
F(x)
means
a ~ b,
-~ CI-
= xk
satisfies
that
one
has
there
the
exists
h
242
[1 ~ h' h' (a) (b)
As borhood ql'
of
-< 2,
zero
0 < ql
is
~ q0'
to
prove
the
equivalent
such
that
to
for
condition
prove
all
F
that
a,
b
for
there
(
f
in
exist
[ - q l q l ],
some
neigh-
C2 > 0
a ~ b
one
and has
Tf (a,b) (7.4.)
RF(a,b ) { C 2 .
Tf(a,b) _ M(a,b) RF~,b) N(a,b) N(a,b)
=
where
(ak-bk)h'
(b)
= ka k - I [h (a) - h (b) ] - ( a k - b k ) h ' (a) (k_l)a k _ kbak-i + b k
M(a,b)
- kbk-l[h(a)-h(b)]
(k-l)b k - kab k-I
Instead C 3, a,
of
0 < C3 < ~ b
6
Clearly,
(7.5.)
kak-1 M(a,b)
and
[ - q 2 , q 2 ],
C 3 S M(a,b)
for
proving q2'
a ~ b
implies
h(a)
belonging
to
= ,izlil =
h' (z) ~Ii + 'l(z) h
=
h' (z)
h
(C2([-q0,q0]),
the
open
h' (a)
for
kak-lh =
interval
- bk
- b h'(z) k bk a a - b
there
' (z) kak-1
linking
a
and
b.
Thus,
=
(k_l)a k _ kbak-i
+ bk
z)
(ak-bk) (z-a)
h'
ak - bk h' (a) a - b k bk a a - b
h'(a)
h' (z)
bk
- h' (a)
z - a
7
(k_l)a k _ kbak-I
(ak-bk) (z-a) ( k - l ) a k :- k b a k - I
to
exists
all
ak
h" (z I )
As
that
that
h'(a)
1 + h' (z) belongs
ak - bk a - b k bk a a - b
a
a
= h' (z)
zI
such
prove
has
k
kak-1
where
~ q0
will
(7.4.).
- h(b) a - b
kak-i M(a,b)
we
(7.5.)
kak-1
z
directly
0 < q2 one
= M(b,a).
+ ak
< Cl~ _
=
some
(7.4.)
and
the
open
h"
is
interval
uniformly
+ bk
linking bounded
a on
and the
b.
interval
[-q0,q0 ] .
243
Moreover,
as w e know,
h"(z 1 ) 6 > 0, h' (z)
small
to f i n i s h
the proof
l i m L(a,b) a~0 b÷0 a~b
where
L(a,b)
L e t us
l i m h' (z) = h' (0) a+0 b÷0 is u n i f o r m l y
of
(7.5.),
The use the
=
that
last
Let
z E 7
z.
the
hand,
Y
together
and
real
< 6
to p r o v e
gives
CA,
r o o t of
l i m T(r) r+l
the d e n o m i n a t o r of
2 = k-l"
is b o u n d e d
where
L(a,b)
the
if
straight
exists
with
Zl(Z)
L(a,b).
on e v e r y
interval
Consequently
way
z
the curve
f .
f
proves
tangent of
passing line
Zl(Z)
only
for
7
one
~2(z),
or
point can always corres-
in a n e i g h -
of a s m o o t h x a 0
at p o i n t £2(z)
through
(x,y)
•
curve.
Let
Z2(z)
and
graph
to
7).
This
(7.6.).
plane
of c o - o r d i n a t e s
is t h e
]r I ~ A 0,
strictly
x S 0
if
z
say t h a t
T-
Y
z E y of
Zl(Z)
lines Y
if
smooth
ends
straight
is d e f i n e d
is o n e of the e n d s of
some n e i g h b o r h o o d
the
system
straight
(7.7.)
oriented
the
that
IL(a,b) I S 2[b].
the o r i e n t e d
to
z
if a t a n y p o i n t
convex
line orthogonal
in a s u i t a b l e
such
inequality
of
in t h e o r t h o g o n a l
of p o i n t
of
Thus,
0 < A < +~.
then
~ A 0,
denote z
A0 > 0
is o n e
to the o r i e n t e d
=
(7.7.)
~
let
function
and
that
Then,
of t h e n u m e r a t o r
function,
if
tangent
ponding
We will
rule
a strictly
that
z
Ibl
b r = -a •
l e t us n o t e
a root
constant
Thus
denote
Orienting
convex
is a l s o
there
assume
borhood
< @,
IL(a,b) I ~ a C A
(one-sided
denote
and
~, a s a c o n t i n u o u s
inequality
Let
z
r = 1
l~(r) I S 21r I .
7.5.
sufficiently
.
The u n i q u e
by s o m e p o s i t i v e
On the o t h e r then
+ bk
a ~ 0
of t h e d e l ' H o s p i t a l
sup a~0,a~b
lal
it is t h e n s u f f i c i e n t
(ak-b k) (a-b) (k-1)a k _ kbak_l
r = i.
function
[-A,A]
when
for
(7.6.)
suppose
is
bounded
Thus,
= 0
(l-r k) (l-r) = a~(r) k (k-l) - kr + r L(a,b)
> 0.
In p a r t i c u l a r ,
satisfies the
zero
fz(X)
the condition
function
(at t h e e n d s
fz of
F
satisfies Y
one
> 0 with this
for all
x ~ 0.
exponent contition
considers
the
C in
244
respective
one-sided
neighborhoods
of zero).
A billiard of class
Pk' 1 _< k _< co belongs to the class Hk if o r there exists a representation F = (i=iUL i) U (3UIFj).__ and a constant C > 0
such that for all strictly convex boundary pieces
(see Sac.
2) belonging
condition
F
with exponent
The class billiards
to some
H3
Li(l_ 0
The p r o o f d~
and
9.
dependinq easily
has a p u r e l y
a really and
is a c o n s t a n t of L e m m a
formal
geometrical It s h o u l d
tiability
of all
7.3.
one.
only
character The p r o o f
be e m p h a s i z e d the c u r v e s
on the
follows
that
from
unlike of L e m m a this
{L i}
and lsisp
differentiability
7.7.
Proof
of t h e s e
of L e m m a
7.3. k£ ~i
curves
Let
which
7.4.
{F
7.4.
is g i v e n
uses
only
} J 1zjsr
Since
Q
belongs
(4.10.)
whose
for
proof
8
differenof
Lemma
7.3.
for
is
in Secs. C1
instead
C3
that c
'
i~ 1
(4. i0. )
d¢(¢,8) k£ - ed elk I dl
~.
formula
is n e c e s s a r y
us r e c a l l
- ed dI
Lemma
proof
billiard
the
kli E 1 dl
k,
to the c l a s s
P3'
then
from
formula
(4.10.)
the
246
estimation
(7.10.)
obviously
it follows
immediately
follows.
From the rule
that to prove the estimation
(~)' (7.11.)
to show that the absolute values of all the quantities ing table are less that
--, a where rI
on
a > 0
f'g -2 fg' g it suffices
from the follow-
is a constant
depending
only
] ~k I
Sk~ .
~d
--(z)
~d I
(z)
Dk]
This is evident zero. _
~k --
For
Sdl ~61 ~81 • ~ - -
~d --
and for
the formula {L i}
Sd(z )
~@ (z)
~(z)
(4.10.) and
0
[G($1)-G(@)]
~$i _
s1 ~ii { [F(@I)-F(@)]F' (%I)+[G(%I)-G(~)]G' (%1)} Thus,
Skl
[G' (,i) -$@
(4.10.)
such that for all
for (@,8)
de ( No
- G'(¢)].
it folone
247
has:
IF
(~i)
-F'
this we obtain _~(%,$Z 8)
7.8.
now describe
F is n o t
below
these
(~i)
- F(%)I+~
IG(%I)
the Misiurewicz
for
f. of
construct Ca
From
(%) I < rl
In fact, such
of
strictly and
7.9.
Using
exmaples the
remark
convex
that
the
the c o n s t r u c t i o n
functions.
for w h i c h
construction
< r12~ _ rla
= f' (0) = 0
in Sec.
boundary
- G(~) I
example
f(0)
class
to t h i s
-
estimation
a whole
with
we pass
IG
such that
we will
billiards
Before
and
satisfied
provides
functions
convex
required
f ~ C~([-I,I])
condition cribed
< rl
_< .~__rl . IF(%l)
We will
function
the
(#)I
of s t r i c t l y
estimate
that
des-
s o m e of
(7.8.)
if for
fails.
f 6 cl([0,a]),
a > 0, lira inf xf' (x) - 1 x÷0 f (x) then
f
from
can not
(7.1.)
(7.12.) which any
satisfy
when
also
fails
= 0
Precisely,
Notice
to s a t i s f y
increasing
and
the c o n d i t i o n
t = 0.
is s a t i s f i e d .
strictly
f(0)
(7.12.)
convex
for e v e r y
x,
Let
fix two arbitrary
positive
every
such
and
b lim n b - a n÷~ n n
that
such that
-
L e t us c h o o s e
X
= 0
{0}
for
U
x
except
F.
described
for
On the o t h e r
f 6 cl([0,a])
one
here,
function
(7.3.)
hand,
for
such that
has
sequences,
{a n } n~l
lima n = limb n÷ ~ n÷~ a n l i m ~-- = O, o r
n
and
{b n }
, of n>-i
= O,
< bn
an
equivalently
< an-1
that
(7.14.)
an a r b i t r a r y 6 {0}
U
function
U [an,b n] n=l
h
and
If s u c h a f u n c t i o n
g 6 C~([0,1])
perhaps
for
n
1
U [an,bn]. n=l
find a function x ~ 0
true
immediately
(7.13.)
n --'-ce
h(x)
examples
is n o t
function
follows
1
numbers,
n ~ 2
for t h e
this
0 < x ~ a,
>
us
This
the condition
xf' (x) f(x)
-
that
F.
an at m o s t
such
that
countable
E C
([0,i])
h(x)
h
> 0
= 0
that
then
one can
for
is fixed,
g(0)
such
and
set of n u m b e r s ,
g(x) and
> 0 that
if
248
I~
ng(t)dt
lim n÷~ I b n h ( t ) d t J0 L e t us d e f i n e Let us d e f i n e
-
Let us fix s u c h a f u n c t i o n
0.
;x
n o w the s e c o n d d e r i v a t i v e
f' (x) =
f"(t)dt
and
f(x)
f"
f (C~([0,1]),
of
=
f
hy
f' [tldt.
0 that
g.
f" = h + g. It is c l e a r
0
that
f(0)
= 0, a n d that
f
is s t r i c t l y
convex.
We w i l l p r o v e n o w that b n f ' ( b n) l im n÷~ This
-
implies
dition
F
(7.15.)
that the f u n c t i o n
f
so d e f i n e d
does not
satisfy
the con-
H' (tldt,
G' (x) =
(tldt
(see 7.12.).
Let us d e f i n e and
i.
f(bn)
G(x)
=
H' (x) =
;x
h(t)dt,
H(x)
=
0
G' (t)dt.
0
Remark
that
H
and
G
are c o n v e x
on the inter -
0 val
[0,i]
and
bnH' (b n) lim n÷~
-
1
7.16.)
H(bn)
Indeed,
by v i r t u e
diately
that for e v e r y
of the d e f i n i t i o n n ~- 1
H(b n) > I bn H' ( t ) d t
=
of f u n c t i o n
h, one d e d u c e s
imme-
one has
(bn-a n) g' (b n)
7.17.)
a n
because
the f u n c t i o n
sequently, (7.14.).
H'
bn H ' ( b n) H(bn )
bn bn _ an
E
Unfortunately,
not s t r i c t l y
convex
is c o n s t a n t
although
on the i n t e r v a l
and thus the
(7.16.)
function
in any n e i g h b o r h o o d
follows
H
(7.15.).
1 E
it is s u f f i c e
lim
to p r o v e
b n G ' ( b n) H(bn )
= 0.
By v i r t u e
of
it is
bnH' (bn)+bnG' (bn)
0
for any
in
M
one
has
and
PART
and
a number
Isin
P =
(0,0)
us
in the
~
(
by
81n
easy
to v e r i f y
formula
such
every
inequality
neighborhood
that
C
that
(4.10.)
this
Consequently,
fails.
ELEMENTARY
following
that
for all
of
M,
element bp
CONFIGURATIONS
sections
the
elements
of
A~
A9
> 0
is a c o m p a c t imbedding.
subset w
of
E Q
P
exists
such
that
then
one has Lemma
there
This
N~,
to p r o v e
7.4.
for all
subset
of
M
and
immediately there
exists
a
r I ~ bQp(W,A~). it s u f f i c e s
its n e i g h b o r h o o d points
w
~ U(P)
As
to p r o v e U(P) D N~
(8.1.) (8.1.)
is not
evident
only
in the c a s e
when
for
U(P)
inf w6U(P)nN~
el(W )
The p r o o f
of L e m m a
consists
and
angles
r I a bpp(W,A~) The
Y
with
note
of the
(7.8.)
ONE:
billiard
8nl'IId~(P,Sn) l; = + ~.
homeomorphic
subset
singular
Let
It is v e r y
is a c o m p a c t
such
Let
elements.
is a
is a c o m p a c t
6 F.
estimate
7.4.
Q
f.
convex
In v i r t u e
lim
NQ = M\Ag,
~ : N~ ÷ M that
A~
the
singular
that
implies
y
n ~ i.
that
In the p r e s e n t
will
strictly
that
(Rn,8~),
implies
function
~ l.
~(P,8 n ) 81 I - + ~. Isin n
8nl
immediately
n
n o w an a r b i t r a r y
F = F1
Isin
corresponding
( y,
=
0
(8.2.)
7.4.
of a d e s c r i p t i o n
consists
of all
of t h r e e
"elementary
steps.
The
first
configurations"
step
of b o u n d a r y
250
pieces The
for w h i c h
"elementary
pose
(8.2.)
holds;
configurations"
any configuration
the r e d u c t i o n asserts
the v a l i d i t y
that
billiards
8.2.
of c l a s s
~ M
i
or
then j
plane
by
F
8.1.
is u s e d
canonical
either
w =
(v,8)
a n d by d e f i n i t i o n
section
we will
6 M L
and
or
=1 v.
vI
always
geometrically
distinct
and of
singular
elements
(z,G)
consider
in d e t a i l
in w h i c h
(8.2.)
(a) P =
z
belongs
(z,0)
or
(b)
z
P =
following
such that
z
z
to a s t r i c t l y
is t h e i r
is the b e g i n n i n g
X-
the
in
(a),
tangent
(b) a n d
(b)
If
then
in t h e
(Fig.
22). in the
B 1
B9
cases
of
y
at
X\{z}
noted
A I,
z
23)
Z(wfl
Indeed, for
if
some
denotes
(see Sec.
the
3.3.).
of b o u n d a r y
(8.2.)
holds,
exhaust
boundary
boundary such
T\{z}
pieces
we will
all
cases
piece
L
passing
~
and
pieces
Yl
T
and
piece
Which
T1
do exist.
T
and
z
is d i s j o i n t z
the b i l l i a r d
and
billiard
Tl\{Z}
through
yl\{~}
y
that the to
boundary
piece
to
and
z
is from
is
trajectories
do exist.
description
is p r e s e n t e d
neighborhood
convex
is the p o i n t
if (Fig.
various 71,
E MF.
of all
small
T
and
24)
cases
o n Fig.
of d i f f e r e n t i a b i l i t y
small
sufficiently
C1 ~ C5
and
Y1
of
is a s t r i c t l y
(Fig.
These
line
is a p o i n t
If
then -
to
sufficiently
y U T!
F.
onto
9.
the
cases
occurring
(c) r e s p e c t i v e l y ,
T h i s case,
when
M
which
and
of
of a c l o s e d
n o w to t h e d e t a i l e d
(a)
z
of
8.1.
that
closed
point
points
boundary
straight
from all points
L e t us p a s s
of Sec.
= z.
cases
convex
only common
of the c l o s e d
going
last
of L e m m a
(v,~)
for w h i c h
of two c l o s e d
from all
the b e g i n n i n g
unilaterally
The
(z,z).
starting
Moreover,
8.4.
configurations
three
of
which
place.
is the b e g i n n i n g
trajectories (c)
the
takes
com-
for the e l e m e n t a r y
is d e f i n e d
~(P)
f i n d all
P =
w =
vI
denote
To
of
Recall
if
8.1.
is the c o n t e n t
in the p r o o f
8.3.
can
in the d e f i n i t i o n
projection
z(w)
v
this
one
of L e m m a
(8.1.)
and
step consists
in Sec.
appears
only
the
between
and
which
8.2.
of w h i c h
The n e x t
inequality
~
distance
In t h i s
H
bricks
pieces.
is d e s c r i b e d
Lemma
the c o n d i t i o n
Denote
w
reduction
of p r o v i n g
like
in Secs.
to the d e m o n s t r a t i o n
of the m a i n
This
step consists
7.4.
is g i v e n
are
of b o u n d a r y
of L e m m a
configuration.
Notice
this
if
and by their
of
arc
z, e x c e p t
we h a v e
T
are and
~i
are
at
tangent
respective
position.
A2 - A5 of arc
z, w e h a v e
transversal
by t h e i r
relative
of
y U T I, for t h e c a s e
the cases
of n o n - d i f f e r e n t i a b i l i t y neighborhood
T1
differ
arc,
22.
of t h e
z, a n d at
T U TI,
cases the
z.
type of convexity
251
(c) and
We can classify
71
for w h i c h
taking well we
all
(8.2.)
into account
the mutual
a s the t r a n s v e r s a l i t y
restrict
cases
ourselves
D 1 - D4,
the configurations
holds
in a s i m i l a r disposition
or tangency
to s u f f i c i e n t l y
E1 - E4
and
of
of
in
L, of
~
and
small
F1 - F3
of b o u n d a r y
w a y as
~
y
L
and
at
and
represented
pieces
7
(b) b u t a l s o y1,
z.
YI'
as
Then,
if
we obtain
o n Figs.
the
25 - 27
respectively. To t h i s noted
by
for e v e r y L(v,9 )
l i s t of c o n f i g u r a t i o n s
G:
y
v 6 ~
tions".
Before
r
A1,
C1 P =
out
has a continuous
ing to
A~,
(v,0) to
singular will
8.3.
to
y
qv
ignore
71.
kind.
"elementary
line
configura-
of r a m i f i c a t i o n
if
exists
where
in a d d i t i o n
another
cases
Pv =
singular
in a n y
elements
Pv =
close
In C a s e
families
z, all
C 5 even
of
U(P)
(V'gv)'
to
(V,~v) , exist.
role
pieces.
neighborhood
singular
and
v
where belong-
two of
Finally,
the c o n f i g u r a t i o n
not play any
to t h e
element
of b o u n d a r y
v E ¥, v ~ z
except
to Figs.
notice
elements:
is s u f f i c i e n t l y
A1
and
C1
in o u r c o n s i d e r a t i o n s
z
22 - 27 w h i c h
To e x h a u s t that
or
all y
in t h e s e
and
is a p o i n t
show only
of
boundary
The c o r r e s p o n d i n g
pieces
always
of
are
these
and we
pieces
y
and
restrict
are
on t h e
A1
ramification
z,
relative
one
posi-
going
the boundary
from
pieces
of
defines
of the b o u n d a r y , 71
which
ourselves
are
to s u f f i -
all
boundary
pieces
y
either
strictly
convex
or
t y p e of c o n v e x i t y
appear
the
trajectories
in c a s e
a n d of
neighborhoods
intervals.
clearly
71
As we z
give
possible
an e x c e p t i o n ,
z
neighborhoods
laying
out that
singular
for o u r c o n s i d e r a t i o n s .
small
be a p o i n t
is s a t i s f i e d .
(v,z)
suppose
22 - 27 w e
will
half
begin.
two c o n t i n u o u s
To a v o i d
o n Figs.
Y1
the
de-
that
them.
Even
and
=
will
Y1 = 7-
ciently
one,
such
can
is s u f f i c i e n t l y
and
Nevertheless,
YI' w e w i l l
essential
the
configurations
many
of
(8.2.)
and
z
there
v
(V'gv)
contain
and
second
pieces
configurations,
pieces
these
and
L e t us t u r n n o w
t i o n of
the
Pv =
elements
simply
following
the e l e m e n t a r y
it t u r n s
in g r e a t
also
z.
C2
(z,0)
such that
families,
out that
family
v ~ z = ~(P)
close
above
l i s t of
boundary
with
that
v 6 Y,
qv =
the
boundary
v I = V l ( V , G ) 6 YI'
listed
to t h e
and
connected
We point
U(P)
add
YI"
36 c a s e s
various
element
(z,z)
that
we will
two disjoint
such that
let us p o i n t
In c a s e s
such
9 to
we pass
in w h i c h
singular
the
are
a few remarks.
Firstly,
one
and
call
let us m a k e
Pl =
Yl
is t r a n s v e r s a l
We will
of
and
figures
of t h e s e and will
boundary in p r i n c i p l e
252
not be d e s c r i b e d The a r r o w s pieces
y
is not
and
shown y
in such
a case
(8.1.)
Y1
22 - 27 show
fixed
by us.
on a s e p a r a t e
changing
as on the
separately.
on Figs.
into
Yl
figure.
and
Y1
It is c l e a r
is c o m p l e t e l y
If we c o n s i d e r
figure
the o r i e n t a t i o n
the o r i e n t a t i o n
but
into
from
y, we w i l l
always
independent
the
from
pieces
truthfulness
the c h o s e n
boundary
a configuration
is o b t a i n e d
of the b o u n d a r y that
of the
a given assume
remain of the
which one
by
that
the
same
inequality
orientation
of
7
and
YI" We w i l l and
z
tions
by
£
and
£i
the o r i e n t e d
to the c u r v e s
denote
y
and
Y1
respectively.
we w i l l
denote
elements
without
singular
elements
With
any
systems
figurations following system,
is the
interval;
note
if
(z, @)y =
=(z'8)F'3 '
(see Sec. pieces
y
z
and
except
is a p o i n t z, y
(z'~)L i , L (z,9)y
L(z,8) 7 = L(z,@)r. 3
the
captions
associate
piece
y
and
In all
71
verified.
are
if
function
is a l s o
to some
y c F..]
the
if
co-ordinate
that
when
L. or i y c L.l
con-
the
co-ordinate
of c l a s s
true
straight
elementary
so small
for the c a s e
= L(z,@)L i
only
the
In the d e s c r i b e d
same
z
capsingular
to the
the
of n o n d i f f e r e n t i a b i l i t y belongs
figure
described.
one c a n
some m o n o t o n i c
moreover,
of
In t h e s e
7.5.).
at p o i n t s
In the
the c o r r e s p o n d i n g
explicitly
to the b o u n d a r y
of
Pv
comment. are
to the y-axis,
neighborhood
and
Pv
configuration
is a l w a y s
graph
belonging
Moreover,
P1
i2(z)
the b o u n d a r y
on a c l o s e d
in some
and
related and
assumption
y
interval
P
elementary
£1(z)
P' PI'
any a d d i t i o n a l
(x,y)
lines
by
tangents
C1
Yl for of r.. 3 and
defined is an
~i" r
then we w i l l (z,e)y
=
253
Configurations
Ai)
A1 - A 5
~)
A2)
/ 7]
Fig.
22.
when
z A2
is
an
: y
of
A3
is
the a
arc
y
of
at
P =
but
is
the
convex
not
convex
begins
U YI"
A 2,
strictly y
strictly
which
: Like
is a
end
boundary z
boundary
excluded.
and
piece z
is
piece.
P : which
the
The
( z , 0 ) y ' P1 begins
point
of
case
=
(z,~)
at
z,
Y.
Yl
differentia-
(z,0)y. roles
of
7
and
71
are
reversed.
(z,~) A4
is
: y = 71 at
interval
bility
P =
A1 is
z
the
Y : 7
and
point
inflection. A5
: Like
Yl
of
are
two
boundary
differentiability
P = A4,
(z,~) but
y
and the
the
arc
of
family y U Y1
Configurations
Bi)
z×
0
pieces
such
pieces
that
y
for all
one has Y'YI
rI
=
r l ( w ) >_ b T , Y l P ( W , A Q )
(8.3.)
258
The
Lemma
implies
9.
8.1.
(8.1.).
PROOF
9.1.
From Lemma
8.1.
9.1.
For
and
Y1
exists
and
arguments
U(s)
it f o l l o w s
to p r o v e
singular of
the
w
6 U(s)
N i
it
s
immediately
that
Lemma.
of b o u n d a r y
pieces
element
and
INEQUALITY
following
s
related
a number
b
- -
for all
to see t h a t
OF THE ~IN
configuration
for e v e r y
a neighborhood
It is e a s y
PROOF
it is s u f f i c i e n t
any elementary
F
9.
for the r e a d e r .
P A R T TWO:
the c o m p a c t n e s s
of
in Sec.
leave details
O F LEM~LA 7.4.
prove
LEMMA
is p r o v e d
We
to
7
to it t h e r e
> 0
such
that
s
one has Y'YI
rI Thus,
=
rl(w)
to p r o v e The
proof
inequality singular ration
t bsP(W,A~) .
the T h e o r e m of L e m m a
(9.1.)
element G
and
related
since
to s t u d y
configurations Sec.
consists
for a n y to
it.
of
to p r o v e
We
have
nothing
denoted
are
cases
it r e m a i n s
In t h i s symbol
for a n y
for c o n f i g u -
particular
respectively,
by t h e i r
the
and
to p r o v e
E1 - E4
9.1.
that
configuration
31 c o n f i g u r a t i o n s .
be s i m p l y
only Lemma
the v e r i f i c a t i o n
elementary
A 2, A 3, B 4 a n d B 5
the remaining will
it r e m a i n s
the c o n f i g u r a t i o n s
of the c o n f i g u r a t i o n s only
7.1.
9.1.
is t r u e
(9.1.)
section
these
introduced
in
8.3. In fact,
few elementary proofs lated
we will
prove
separately
configurations
are analogous
or t h e y
because follow
the
inequality
for t h e o t h e r
from
(9.1.)
ones
only
either
the Proposition
9.1.
for a
the
formu-
below. All
the p r o o f s
condition
F,
them.
Notice
in
of
M
proved. denoted
only
a r e of v e r y the
that we will
singular
by the
same
define
for w h i c h
element
symbol
nature
geometrical
never
elements
For a singular
elementary
simplest
s
U(s).
explicitly
the e s t i m a t e
all
these
and,
except
considerations
in
the neighborhoods (9.1.)
will
neighborhoods
Nonetheless,
for the
are used
this will
be
will never
be lead
to
ambiguity.
9.2.
First
PROPOSITION mentary for a n y
of all
9.1.
let us p r o v e
Let
the b o u n d a r y
configuration. singular
the
element
If
Y1 Pv =
following
pieces
seen (V'ev)
from
y 7
6 fy,yl
proposition.
and
Y1
form an ele-
is s t r i c t l y
concave,
NAq 4
is n o t
which
an
then
259
isolated bp
point
of
4
[
N A~,
the
inequality
(9.1.)
holds
and
rigorously
with
Y'YI
= 1/2. v
Proof.
It is g e o m e t r i c a l l y
in the p r o o f v
on
that
7 P~v =
function
of T h e o r e m
such
of
that
(~,@v) ~
evident
6.1
that
for e v e r y
( t Y'71 (see Fig.
N AQ4
there
Q 6 Uv and
28 w h e r e
it was exists there
such y
that
demonstrated
a neighborhood exists @v~
is s t r i c t l y
an a n g l e
Uv
of
@~
such
is a c o n t i n u o u s convex).
260
Fig.
28.
Let has
not
!@v -@I,
us d e n o t e
that
only
y
6 :
sin
(v,0).
from
w =
p(w,A~)
6
0 < 61 < 7/2,
S p(w,P~)
is s u f f i c i e n t l y
depend
on
Notice
that
and
Y1
This
v E U
V
=
9.3.
A 1 - In this
7
one
has
proof
the
ends
of
y
of
y
only
it is s u f f i c i e n t neighborhoods that one
case
singular
it e a s i l y
one
81(w) I, if
where
60
does
v
is a c o m m o n
point
of
the v a l i d i t y
D 2, F 1 and F 2.
The
of L e m m a
remaining
7.4.
twenty
two
separately.
for any
point
elements, such
element
that
to p r o v e
to p r o v e
that
and (u,Q)
for
U(qv)
v
qv =
one
w =
when
implies
follows
U(qv)
for e v e r y
the c a s e
immediately
be c o n s i d e r e d
two
61 = 2 1 s i n
0 < 6 ~ 60
Y1
•
for B 4, B 7, B 9, C I, C 3, C5, will
if
of
excluded.
proposition
cases
i.e.
@l(W) I.
the c o n c a v i t y
6 S 61 ~ 2 sin
small,
Isin
.
in this
is not
Thus,
61 =
exists. the
every in
E U(qv ) n
belonging (v,0)y
M
From
point
and
of At
the c o m p a c t n e s s (9.1.)
v E 7
a number w =
interior
q v .= (v,~)y.
inequality
and
iy,y
to the
and
there bv > 0
(u,Q)
for
A1
exist such
E U(qv) A [ Y'YI
has rI
:
r l(w)
-> b v P ( W , A g )
rl
:
rl (~)
>- bvP (w,Ag)
As
qv
(9.2.)
and
and
q5
play
a completely
symmetrical
role
in the a b o v e
261
inequalities,
it is s u f f i c i e n t
To p r o v e Cv > 0 w =
and
(u,0) tan
(9.2.)
it is e n o u g h
a sufficiently
6 U(qv)
01(u,9)
to p r o v e
N
i
one
~ C v tan
to s h o w
small
Y,Y
only
(9.2.)
that
there
neighborhood
exists
U(qv)
a number
such
that
for all
has
9
(9.3.) C
From in
(9.3.) U(qv)
one
N
piece
any y
restriction
is so small
In the c o - o r d i n a t e and
Z2(v), Now,
v 6 y
duced
Y
~
related
in Sec.
7.5.
in some
of a s t r i c t l y F.
When
v
one-sided when
Let
w =
f(s)
< f(t),
(d)
s,
to the
(9.2.)
is true
with
some
an end
(u,8) We w i l l (b)
of
small.
bv -
v 2
y,
function
us
line
one
v =
Here
lines
il(V)
C I.
(v,9)
where
Z2(v),
In this
(0,0),
bound-
the c o - o r d i n a t e
and
F.
y
intro-
co-ordinate
is the
satisfying
considers
(0,0).
w =
introduce
g = fv
the
of c l a s s
Zl(V)
the c o n d i t i o n
then
v =
that
is v e r i f i e d .
straight
the p o i n t s
Let
straight
function
of of
convex
Consider
assume
condition
to the
of the p o i n t
smooth
is an end
we can
following related
defining
neighborhood convex
is not
(t,fv(t)).
of
(9.3.).
while
neighborhoods
v
(x,y)
graph
prove
the
is s u f f i c i e n t l y
(x,y)
system
that
of g e n e r a l i t y that
system
is the
we w i l l
and
system
immediately
Ly,y.
Without ary
deduces
graph
the c o n d i t i o n
the c o r r e s p o n d i n g
we w i l l
consider
the c a s e
y.
( /y,y.
Let
distinguish s < 0 ~ t
u = four
and
(S,fv(S)) cases:
f(s)
and (a)
{ f(t),
let
u I = Ul(U,G)
s S 0 < t (c)
and
s, t { O,
t ~ 0.
Consider
first
the c a s e
(a) w h i c h
is p r e s e n t e d
Y
29.
y=g ( x / U
v=(0,0) u
Fig.
on Fig.
29
/
t
1
=
262
Clearly, Thus
tan
+g(t)
@ =
~
+
and
one
considers
To
prove
(9.3.)
to
and
prove
=
tan
only
cient
~
tan(a+6)
- g(s) - s
t if
8 =
s
that
and
the
for
:
1
B-
6.
where
tan
f (t) t
6 =
-
f (s) s
t a n ~ + t a n 6 ~ t a n a + t a n .4 = -g' (s) i-tan ~ tan 6 tan(B-a) : tan 5 - tan ~ 1 _g(t)-g(s)) = 1 T-tan B ~n &-2(g' (t) t - s
@i
in
8
t
sufficiently
case
some
(a)
C
>
under 0
small.
consideration,
and
for
all
s,
it t,
s
is
suffi-
_< 0
< t,
I sJ
V
and
t
sufficiently
g' (t)
- g(t) t g(t) - g(s) ts
However the
(9.4.)
small,
-
g(s) s
g =
has
(9.4.)
v
(s)
states
function
£ C
, g
one
f
nothing .
but
Thus,
the
(9.4.)
validity
is
true
of
and
the
one
condition
can
take
F
as
for
C
the
V
exponent In In
all
y. cases
these
It A1
of the
follows y
A2 -
Let
related
to of
(b),
cases
iff
graph
V
from
the
=
proof
the
(d),
convex
the
to
line
that
the
are
We
completely
omit
the
inequality
analogous.
details.
(8.3.)
holds
for
F.
We
will
£1(z)
smooth
proofs
(9.4.).
condition
(v,8) ( /y,71.
straight
strictly
and
arrives
this
satisfies
w
(c)
one
use
and
function
the
co-ordinate
£2(z). g
For
defined
x on
system £
the
0,
y
(x,y)
is
a
interval
[0,A] .
%
£2 (x)
£
,
z=(0,0)
vI
F i g . 30. parallel
v to
=
(t,g(t)) £1(z).
for
some
t
6
A
[0,A] .
The
straight
line
£1 (z)
i
is
'
263
By virtue defined
on Fig.
On the other Thus,
of t h e d e f i n i t i o n
singular
reasoning
for
A2
elements
will
implies
appear
One
has h e r e
Let
w =
(v,e)
z/2
> el(W) that
> 81
5 A~.
that
in m o s t
{ Cz
b > 0
the
one
where
Y1
(8.3.)
(z,e).
is
a p((v,e),A~).
immediately
follows.
of t h e
modifications
the
use
same
to f o l l o w .
=
(v,~)
is a s t r i c t l y
w =
on
qv
~
0 < 81 < ~/2.
as an e x a m p l e
very minor
elements
tan ~
depends
where
inequality
typical
As
E A 95 c AQ w h e r e
p ( ( v , @ ) , (v,~))
of the p r o o f s
singular
) > 0
(v,~)
= ~ - eI
With
( iy,yl
tan el(W)
where
and
is a v e r y of
A3
has be
This
p ( ( v , e ) , (v,~))
~ - eI { p((v,e),A~)
The p r o o f of
30.
hand
5 A2,
of
where convex
However,
and
in c o n s e q u e n c e
Yl
only.
Thus,
the
then
(9.3.)
from
dl(W)
v 6 y. curve,
= sin
inequality
one
el(W) (8.3.)
is p r o v e d .
A 4 - Let the
w =
first when
(v,e)
E iy,71
0 < e < z
L e t us c o n s i d e r exists
a unique
(9.1.)
then
the
e
and
first
We will the
such that
P
V
follows
~ Fig.
31.
~
=
second
case
when
(see Fig. =
(v,e v)
separately:
~ < e < 2~.
31). ~ A4 n
two cases
In t h i s
case
and
inequality
the
there
V
from Proposition
9.1.
£v
e
consider
(see Fig.
31).
~Y1 ~v~ZI
-
ev ,
0