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Thèse de doctorat/ PhD Thesis Citation APA:

Heistercamp, M. (2011). The Weinstein conjecture with multiplicities on spherizations (Unpublished doctoral dissertation). Université libre de Bruxelles, Faculté des Sciences – Mathématiques, Bruxelles.

Disponible à / Available at permalink : https://dipot.ulb.ac.be/dspace/bitstream/2013/209882/4/86d6018e-71a3-488e-9506-e3eb3cfd7b4a.txt

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D 0 3 8 1 9

U L B

U N I V E R S I TÉ L I B R E

DE B R U X E L L E S

m i i e

UNIVERSITÉ DE NEUCHâTEL

T h e W e i n s t e i n c o n j e c t u r e w i t h m u l t i p H c i t i e s o n s p h e r i z a t i o n s

T h è s e présentée en v u e de l'obtention d u grade d e D o c t e u r en Sciences

p a r

Muriel HEISTERCAMP

D i r e c t e u r s d e t hè s e :

Prof. F. Schlenk, Université de Neuchâtel Prof. F. Bourgeois, Université Libre de Bruxelles C o m p o s i t i o n d u j u r y :

Président: Prof. A. Valette, Université de Neuchâtel Secrétaire: Prof. S. G u t t , Université Libre de Bruxelles Examinateurs: Prof. A. Abbondandolo, Uiiiversità di Pisa

Prof. M. Bertelson, Université Libre de Bruxelles

Année académique 2010-2011

U n i v e r s i t é L i b r e d e B r u x e l l e s

Y

U L B ^LIBRE

DE B R U X E L L E S

m i

UNIVERSITÉ DE NEUCHâTEL

T h e W e i n s t e i n c o n j e c t u r e w i t h m u l t i p l i c i t i e s o n s p h e r i z a t i o n s

T h è s e présentée e n v u e d e l ' o b t e n t i o n d u g r a d e d e D o c t e u r e n Sciences

p a r

Muriel HEISTERCAMP

D i r e c t e u r s d e t h è s e :

Prof. F. Schlenk, Université de Neuchâtel Prof. F. Bourgeois, Université Libre de Bruxelles C o m p o s i t i o n d u j u r y :

Président: Prof A. Valette, Université de Neuchâtel Secrétaire: Prof. S. G u t t , Université Libre de Bruxelles Examinateurs: Prof. A. Abbondandolo, Université di Pisa

Prof. M. Bertelson, Université Libre de Bruxelles

Année académique 2010-2011

À Nelle.

A b s t r a c t i

A c k n o w l e g m e n t s iii I n t r o d u c t i o n V 1 D é f i n i t i o n s a n d T o o l s 1

1.1 T h e free loop s p a c e 1

1.1.1 G r o w t h c o m i n g f r o m C{M) 3 1.1.2 Energy h y p e r b o l i c m a n i f o l d s 4 1.1.3 E x a m p l e s 5

1.2 C o t a n g e n t b u n d l e s 9

1.2.1 Fiberwise s t a r s h a p e d h y p e r s u r f a c e s 10

1.2.2 D y n a m i c s on fiberwise s t a r s h a p e d h y p e r s u r f a c e s 12 1.2.3 Spherization of a c o t a n g e n t b u n d l e 14

1.3 Maslov index 15

1.3.1 Maslov i n d e x for s y m p l e c t i c p a t h 16 1.3.2 T h e Maslov i n d e x for L a g r a n g i a n p a t h s 17 1.3.3 Maslov i n d e x for periodic o r b i t s 17 2 C o n v e x t o S t a r s h a p e d 1 9

2.1 Relevant H a m i l t o n i a n s 19 2.2 A c t i o n spectra 22

2.2.1 T h e Non-crossing l e m m a 26

2 . 3 F l o e r Homology for H a m i l t o n i a n s convex a t infinity 29 2.3.1 Définition of i f F f ( / / ; Fp) 29

2.3.2 C o n t i n u a t i o n h o m o m o r p h i s m s 32

2.4 F r o m Floer h o m o l o g y t o t h e h o m o l o g y of t h e free loop space 34 2.4.1 C o n t i n u a t i o n h o m o m o r p h i s m s 34

2.4.2 To the h o m o l o g y of t h e free loop space 37

v i C o n t e n t s

3 M o r s e - B o t t h o m o l o g y 4 1 3.1 A M o r s e - B o t t s i t u a t i o n 41 3.2 A n a d d i t i o n a l p e r t u r b a t i o n 42 3.3 M o r s e - B o t t homology 44

4 P r o o f s o f t h e o r e m A a n d t h e o r e m B 4 9 4.1 P r o o f of t h e o r e m A 49

4.2 P r o o f of t h e o r e m B 52

4.2.1 T h e simply c o n n e c t e d case: generalizing B a l l m a n n - Z i l l e r . . . 59 5 E v a l u a t i o n 6 3

5.1 Lie g r o u p s 63

5.2 TTi(M) finite: t h e case of t h e s p h è r e 64 5 . 3 N é g a t i v e c u r v a t u r e 67

A C o n v e x i t y 6 9

B L e g e n d r e t r a n s f o r m 7 3 C G r o m o v ' s t h e o r e m 7 5

B i b l i o g r a p h y 7 9

Abstract

L e t M b e a s m o o t h closed m a n i f o l d a n d T*M its c o t a n g e n t b u n d l e endowed w i t h t h e u s u a l symplectic s t r u c t u r e ui = d\, where A is t h e Liouville f o r m . A h y p e r s u r f a c e E C T * M is said to be fiberwise starshaped if for each point q £ M t h e intersection E , : = S n T * M of E w i t h t h e fiber a t q is t h e s m o o t h b o u n d a r y of a d o m a i n in T*M w h i c h is s t a r s h a p e d w i t h r e s p e c t t o t h e origin 0 , eT*M.

In t h i s thesis we give lower b o u n d s on t h e g r o w t h r a t e of t h e n u m b e r of closed R e e b o r b i t s o n a fiberwise starshaped hypersurface in t e r m s of t h e t o p o l o g y of t h e free

l o o p s p a c e of M. We d i s t i n g u i s h t h e t w o cases t h a t t h e f u n d a m e n t a l g r o u p of t h e

b a s e s p a c e M h a s an e x p o n e n t i a l g r o w t h of c o n j u g a c y classes or n o t . If t h e base

s p a c e M is s i m p l y connected we generalize t h e t h e o r e m of B a l l m a n n a n d ZiUer o n

t h e g r o w t h of closed geodesics t o R e e b flows.

Acknowlegments

I w o u l d like t o t h a n k m y advisor Félix Schlenk for his s u p p o r t t r o u g h o u t t h i s work a n d t h e o p p o r t u n i t y t o c o m p l è t e t h i s p r o j e c t . E v e n w h e n a b r o a d , h e never failed t o a n s w e r m y questions a n d prevent m e t o follow w r o n g ideas. D u r i n g ail t h è s e years I e n j o y e d om t h e never e n d i n g m a t h discussions in t r a i n s or over cofïee breaks.

F u t h e r m o r e I t h a n k Frédéric Bourgeois, U r s F r a u e n f e l d e r a n d A l e x a n d r u O a n c e a f o r v a l u a b l e discussions. I also t h a n k Will M e r r y a n d Gabriel P a t e r n a i n for h a v i n g p o i n t e d o u t t o m e t h a t t h e m e t h o d s used in t h e proof of T h e o r e m s A a n d B can a l s o b e used t o prove T h e o r e m C.

J ' a i m e r a i s remercier Agnes Gaxibled p o u r avoir t r o u v e r les p e t i t e s e r r e u r s a u milieu d e m e s interminables exposés et A l e x a n d r e G i r o u a r d p o u r ses conseils d e r é d a c t i o n . J e v o u s n o m m e marraine et p a r r a i n d e c e t t e thèse.

C e t t e t h è s e m ' a u r a fait d é m é n a g e r d e l ' U L B à l'Unine. J ' a i eu la chance d e travailler d e s d e u x côtés entourée d e gens d i s p o n i b l e s et t o u j o u r s p r ê t s à se r e t r o u v e r a u t o u r d ' u n dessert o u d'un a p é r o . P o u r Bruxelles merci à Céline, M a u d e , M a t h i e u , Seb, J u l i e , Nicolas, Michael, A n n et J o ë l . L e d é p a r t n e f u t p a s si facile, o n r e t o u r n e c h e z C a p o u e q u a n d vous voulez. P o u r N e u c h â t e l merci à D o r o t h é e , Olivier, G r e g , B é a t r i c e D., David G., M a r i a , D a v i d F . , R a p h a ë l , Denis et Alain, l'accueil f u t im- p e c c a b l e .

Lise e t Kola, chacun à v o t r e m a n i è r e v o u s m ' a u r e z a p p o r t e r ce s o u t i e n et c e t t e s é r é n i t é quotidienne qui fait q u ' o n est h e u r e u x de rentrer chez soi. J e v o u s e n re- m e r c i e , v o u s faites partie d e la famille m a i n t e n a n t .

M e r c i à t o u s ceux cher à m o n c o e u r q u i m ' o n t suivit, à d é f a u t d u b o u t d u m o n d e ,

j u s q u ' e n Suisse, Sylvie, A r i a n e , M a g a l i , M a ï t é , Vanessa, D a p h n é , M a t h i e u , Virginie,

F a b i a n , Audrey, Marine, Olivier, M a r i e , Marie, D j u , Viri, Florence, Coralie et les

p o m m e s . Vous m'avez bien aidée à g a r d e r u n pied en Belgique et la t ê t e en Suisse.

Merci à B é a t r i c e H. p o u r avoir organisé le p r e m i e r voyage d a n s l ' a u t r e sens, j ' e s p è r e qu'il y e n a u r a d ' a u t r e s !

Régis t u a s réussi à r e n d r e le l u n d i m o n j o u r préféré d e la semaine! Merci p o u r t o n soutien et t a présence qui o n t bien a d o u c i la p é r i o d e d e r é d a c t i o n .

F i n a l e m e n t merci à mes p a r e n t s , j e vous dois é n o r m é m e n t . Merci p o u r v o t r e s o u t i e n

s a n s faille, v o t r e présence, vos conseils. J e n ' y serais p a s arriver s a n s vous.

Introduction

Let M b e a s m o o t h closed m a n i f o l d a n d d é n o t e b y T*M t h e c o t a n g e n t b u n d l e over M e n d o w e d w i t h its usual s y m p l e c t i c s t r u c t u r e us = d\ where X = pdq = Y^^i Pi is t h e Liouville form. A h y p e r s u r f a c e S C T*M is said t o b e fiherwise starshaped if for e a c h p o i n t q £ M t h e intersection S , : = E n T*M of S w i t h t h e fiber a t q is t h e s m o o t h b o u n d a r y of a d o m a i n s t a r s h a p e d w i t h respect t o t h e origin 0 , G T*M.

T h e r e is a flow naturally a s s o c i a t e d t o E , g e n e r a t e d b y t h e u n i q u e vector field R a l o n g E defined by

dX{R,-)=0, \{R) = l.

T h e v e c t o r field R is called t h e Reeb vector field o n E a n d its flow is called t h e Reeb flow. T h e m a i n resuit of t h i s thesis is t o prove t h a t t h e topological s t r u c t u r e of M forces, for ail fiberwise s t a r s h a p e d h y p e r s u r f a c e s E, t h e existence of m a n y closed o r b i t s of t h e R e e b flow o n E . M o r e precisely, we shall give a lower b o u n d of t h e g r o w t h r a t e w i t h respect t o t h e p e r i o d s of t h e n u m b e r of closed R e e b - o r b i t s in t e r m s of t h e t o p o l o g y of the m a n i f o l d .

T h e e x i s t e n c e of one closed o r b i t w a s c o n j e c t u r e d by Weinstein in 1978 in a m o r e gênerai s e t t i n g .

W e i n s t e i n c o n j e c t u r e . A hypersurface E of contact type and satisfying H^{11) = 0 carries a closed characteristic.

I n d e p e n d e n t l y , Weinstein [49] a n d R a b i n o w i t z [38] established t h e existence of a closed o r b i t o n star-like h y p e r s u r f a c e s in IR^". In our s e t t i n g t h e W e i n s t e i n conjec- t u r e w i t h o u t t h e a s s u m p t i o n i ï H ^ ) = 0 was proved in 1988 by Hofer a n d V i t e r b o , [26]. T h e existence of m a n y closed o r b i t s h a s already b e e n well s t u d i e d in t h e spé- cial case of the géodésie flow, for e x a m p l e by G r o m o v [24], P a t e r n a i n [34, 35] a n d P a t e r n a i n - P e t e a n [37]. In t h i s thesis we will generalize their results.

T h e p r o b l e m a t h a n d c a n b e considered in two équivalent ways. F i r s t , let H: T*M ^

IR b e a s m o o t h H a m i l t o n i a n f u n c t i o n s u c h t h a t E is a regular level of H. T h e n t h e

H a m i l t o n i a n flow ifn of H is o r b i t - e q u i v a l e n t t o t h e R e e b flow. T h e r e f o r e , t h e

g r o w t h of closed o r b i t s of ipn equals t h e g r o w t h of closed o r b i t s of t h e R e e b fiow.

Secondly, let SM b e t h e oosphère b u n d l e w i t h respect t o a choosen R i e m a n n i a n m e t r i c over M endowed w i t h its canonical c o n t a c t s t r u c t u r e ^ = ker A. T h e c o n t a c t manifold (SM,^) is called t h e s p h e r i z a t i o n of M. O u r m a i n results are équivalent t o saying t h a t for any c o n t a c t f o r m a for ^, i.e. ^ = ker a , t h e g r o w t h r a t e of t h e n u m b e r of closed o r b i t s of t h e R e e b fiow of a in t e r m s of their p e r i o d d é p e n d s only on M a n d is b o u n d e d f r o m below by homological d a t a of M.

T h e f r e e l o o p s p a c e

In t h e following we use t h e définitions a n c o n c e p t s of [31] i n t r o d u c e d t o s t u d y t h e oomplexity of t h e based loop s p a c e a n d a d a p t t h e m t o t h e free loop space. T h e complexity of t h e R e e b fiow o n E C T*M cornes f r o m t h e oomplexity of t h e free loop s p a c e of t h e b a s e manifold M. Let ( M , g) b e a C ° ° - s m o o t h , closed, oonnected R i e m a n n i a n m a n i f o l d . Let A M b e t h e free loop s p a c e of M, i.e. t h e set of loops q : M oî Sobolcv class W^''^. T h i s s p a c e h a s a canonical Hilbert m a n i f o l d s t r u c t u r e , see [28]. T h e energy f u n c t i o n a l £ = £g : KM ^ IR is defined by

w h e r e |g(^)|^ = 9q{t){Q{t),q{t)). For a > 0 we oonsider t h e sublevel sets A" : = {qeAM\ £{q) < a}.

Now let Po b e t h e set of p r i m e n u m b e r s a n d w r i t e P : = PQ U {0}. For each p r i m e n u m b e r p d é n o t e by Fp t h e field Z / p Z , a n d w r i t e FQ : = Q . T h r o u g h o u t , Ht will d é n o t e singular homology a n d

t f c : H f c ( A ' ' ; F p ) ^ H f c ( A M ; F p )

t h e h o m o m o r p h i s m induced by t h e inclusion A " M ^ A M . Following [20] we m a k e t h e

D é f i n i t i o n . The Riemannian manifold {M, g) is energy hyperbolic if

C(M,g) : = s u p l i m i n f - l o g V d i m t f c / f f c ( A 5 " ' ; F p ) > 0.

Remark. T h e choice of t h e sublevel sets in t h e définition m i g h t seems not n a t u r a l for t h e r e a d e r . It is induced by t h e fact t h a t for geodesics H a m i t o n i a n flows, n - p e r i o d i c s o r b i t s c o r r e p o n d s t o loops of energy ^n"^.

Since M is closed, t h e p r o p e r t y energy hyperbolic d o e s n o t d é p e n d on g while, of

course, C{M,g) d o e s d é p e n d on g. W e say t h a t t h e closed manifold M is energy

I n t r o d u c t i o n v i i hyperbolic if {M, g) is e n e r g y h y p e r b o l i c for s o m e a n d hence for a n y R i e m a n n i a n

m e t r i c g o n M . For ail o > 0 we h a v e C{M,ag) = -^C{M,g).

W e also consider t h e polynomial growth of t h e h o m o l o g y given by

c{M,g) : = s u p l i m i n f - ^ l o g V d i m i f c F f c ( A ^ " ' ; F p ) . pgp n^co l o g n ^

D é n o t e by A ^ M t h e c o n n e c t e d c o m p o n e n t of a loop a in A M a n d by A i M t h e c o m p o n e n t of contractible loops. T h e c o m p o n e n t s of t h e loop space AM a r e in b i j e c t i o n w i t h t h e set C ( M ) of c o n j u g a c y classes in t h e f u n d a m e n t a l g r o u p 7ri(M), w e c a n t h u s d é n o t e by Ac t h e c o n n e c t e d c o m p o n e n t of t h e é l é m e n t s of t h e class c i.e.

A M = ]J A c M .

c e C ( M )

F o r e a c h é l é m e n t c e C{M) d é n o t e by e(c) t h e i n f i m u m of t h e energy of a closed cmve r e p r e s e n t i n g c. Let C ° ( M ) : = {c € C{M) \ e(c) < a}, a n d define

E{M) •- l i m i n f - l o g # C ° ( M ) , a—>oo a

e ( M ) : = l i m i n f - ^ l o g # C " ( M ) . a->oo log a

N o t e t h a t E{M) has t h e s a m e d e p e n d e n c e on t h e m e t r i c g as t h e one of C{M,g), moieovev C {M, g) > E{M).

F i b e r w i s e s t a r s h a p e d h y p e r s u r f a c e s i n T*M

T h e foUowing définition cornes f r o m [31]. Let S b e a s m o o t h c o n n e c t e d h y p e r s u r f a c e i n T*M. W e s a y t h a t S is fiberwise starshaped if for each point q E M t h e set

: = E n T * M is t h e s m o o t h b o u n d a r y of a d o m a i n in T*M w h i c h is s t r i c t l y s t a r s h a p e d w i t h respect t o t h e origin 0 , € T*M. T h i s m e a n s t h a t t h e r a d i a l vector field YliPi transverse t o each E , . W e a s s u m e t h r o u g h o u t t h a t d i m M > 2.

T h e n T * M \ E h a s two c o m p o n e n t s , t h e b o u n d e d inner p a r t -D(E) c o n t a i n i n g t h e z é r o section a n d t h e u n b o u n d e d o u t e r p a r t D'^ÇS) = T*M \ -D(E), w h e r e D(E)

d é n o t e s t h e closure of D{Y,).

F o r m u l a t i o n o f t h e r e s u l t s

L e t E C T*M b e as above a n d d é n o t e b y ( f R t h e R e e b flow on S . A closed o r b i t 7

of t h e R e e b flow will b e said simple if it is geometrically différent f r o m ail t h e o t h e r

o r b i t s , i.e. if 7 7^ 7 * for ail 7 ' closed R e e b o r b i t , /c e N. For r > 0 let ORÇT) b e t h e

s e t of simple closed o r b i t s of (p^ w i t h p e r i o d < T. W e m e a s u r e t h e g r o w t h of t h e

n u m b e r of é l é m e n t s in OJI{T) by

NR : = l i m i n f - l o g ( # C » H ( T ) ) , T—^OO T

UR : = l i m i n f - ^ l o g ( # O H ( r ) ) . T-KX> l O g T

T h e n u m b e r NR is t h e exponential growth rate of closed o r b i t s , while TIR is t h e polynomial growth rate. T h e foUowing t h r e e t h e o r e m s a r e t h e m a i n r e s u i t of t h i s thesis.

T h e o r e m A . Let M be a closed, connected, orientable, smooth manifold and let E C T*M be a fiberwise starshaped hypersurface. Let ( f R be the Reeb flow on E , and let NR, UR, E{M) and e ( M ) he defined as above. Then

(i) NR > E { M ) ; (a) UR > e{M) - 1.

W e will say t h a t E is generic if e a c h closed R e e b o r b i t is t r a n s v e r s a l l y n o n d e g e n e r a t e , i.e.

d e t ( / - d ^ ^ ( 7 ( 0 ) ) | ç ) ^ 0 . T h e o r e m B . Let M be a closed, connected, orientable smooth manifold and let E C T'M be a generic fiberwise starshaped hypersurface. Let ( f R be the Reeb flow on E , and let NR, UR, C{M, g) and c{M, g) be defined as above. Then

(i) NR>C{M,g).

(ii) UR > c{M,g) - 1.

T h e h y p o t h e s l s of genericity of E allow u s t o achieve a M o r s e - B o t t s i t u a t i o n in t h e foUowing way: t h e H a m i l t o n i a n f u n c t i o n t h a t we will consider a r e a u t o n o m o u s . T h i s implies t h a t t h e closed o r b i t s of t h e i r H a m i l t o n i a n flow a r e d e g e n e r a t e d a t least in t h e d i r e c t i o n of E , i.e. 1 is a n eigenvalue of t h e t i m e - l - r e t u r n m a p of t h e flow. W e t h u s a s k t h i s d i r e c t i o n t o b e t h e o n l y one.

T h e i d e a of t h e p r o o f s is a s foUows. L e t E c T*M b e a fiberwise s t a r s h a p e d h y p e r s u r f a c e . If E is t h e level set of a H a m i l t o n i a n f u n c t i o n F : T'M —> H , t h e n t h e R e e b flow of A is a r e p a r a m e t r i z a t i o n of t h e H a m i l t o n i a n flow. W e c a n d e f i n e such a H a m i l t o n i a n b y t h e two c o n d i t i o n s

F\^ = l , F{q,sp) = s^F{q,p), s > 0 smd {q,p) e T*M. (1)

T h i s H a m i l t o n i a n is n o t s m o o t h n e a r t h e zéro s e c t i o n , we t h u s d e f i n e a cut-off f u n c - tion / t o o b t a i n a s m o o t h f u n c t i o n f o F. W e t h e n use t h e i d e a of s a n d w i c h i n g

I n t r o d u c t i o n î x

d e v e l o p e d in P r a u e n f e l d e r - S c h l e n k [19] a n d M a c a r i n i - S c h l e n k [31]. B y s a n d w l c h - i n g t h e set S between t h e levai s e t s of a géodésie H a m i l t o n i a n , a n d b y using t h e H a m i l t o n i a n F l o e r homology a n d its i s o m o r p h i s m t o t h e h o m o l o g y of t h e free loop s p a c e of M, w e shall s h o w t h a t t h e n u m b e r of l - p e r i o d i c o r b i t s of F of a c t i o n < a is b o u n d e d b e l o w by t h e r a n k of t h e h o m o m o r p h i s m

t f c : i / f c ( A " ' ; F p ) - > i f f c ( A M ; F p ) i n d u c e d by t h e inclusion A"'M ^ KM.

S i n c e F is a u t o n o m o u s , ail i t s p e r i o d i c o r b i t s a r e d e g e n e r a t e in a t least o n e d i r e c t i o n . W e t h u s n e e d t o consider s m a l l t i m e - d e p e n d a n t p e r t u b a t i o n s of F. In t h e proof of T h e o r e m A, we will a d d t o F s m a l l p o t e n t i a l s of t h e f o r m Vi{t,q). A s s u m i n g

||Vi(f,9)||c«> —>• 0 for / —>^ oo, we will s h o w t h e e x i s t e n c e of a p e r i o d i c o r b i t of F i n e v e r y non-trivial c o n j u g a c y class a s t h e limit of p e r i o d i c o r b i t s of F + VJ.

T h i s s t r a t e g y cannot b e a p p l i e d for T h e o r e m B. W e t h u s use t h e a s s u m p t i o n of g e n e r i c i t y t o achieve a M o r s e - B o t t s i t u a t i o n following P r a u e n f e l d e r [18, A p p e n d i x A] a n d B o u r g e o i s - O a n c e a [7] a n d u s e t h e Correspondence Theorem b e t w e e n M o r s e h o m o l o g y a n d Floer h o m o l o g y d u e t o B o u r g e o i s - O a n c e a , [7], t o o b t a i n o u r r e s u i t . Remark. A p r o o f of rough versions of T h e o r e m s A a n d B is o u t l i n e d in Section 4 a of S e i d e l ' s s u r v e y [44]. M e a n w h i l e , a d i f f é r e n t ( a n d difiicult) proof of t h è s e t h e o r e m s , w i t h coefficients in Z2 only, w a s given by M a c a r i n i - M e r r y - P a t e r n a i n in [30], w h e r e a v e r s i o n of R a b i n o w i t z - F l o e r h o m o l o g y is c o n t r u c t e d t o give lower b o u n d s for t h e g r o w t h r a t e of leaf-wise i n t e r s e c t i o n s .

S p h e r i z a t i o n o f a c o t c i n g e n t b u n d i e

T h e h y p e r p l a n e field = ker <z T S is a c o n t a c t s t r u c t u r e o n E . If E ' is a n o t h e r f i b e r w i s e s t a r s h a p e d h y p e r s u r f a c e , t h e n ( E , ^ e ) a n d {T,',^^i) a r e c o n t a c t o m o r p h i c . I n f a c t t h e difïerential of t h e d i f î e o m o r p h i s m o b t a i n e d by t h e r a d i a l p r o j e c t i o n m a p s

^s'- T h e équivalence class of t h è s e c o n t a c t m a n i f o l d s is called t h e spherization {SM,^) of t h e cotangent b u n d i e (T'MJLJ). T h e o r e m A a n d T h e o r e m B gives lower b o u n d s of t h e growth r a t e of closed o r b i t s for any Reeb flow on the spherization SM ofT*M.

S p é c i a l e x a m p l e s of fiberwise s t a r s h a p e d h y p e r s u r f a c e s a r e u n i t c o s p h e r e b u n d l e s Si M (g) a s s o c i a t e d to a R i e m m a n i a n m e t r i c g,

SiMig) := {{q,p)eT*M\\p\ = l}.

T h e R e e b flow is then t h e géodésie flow. In t h i s case, T h e o r e m A is a direct con- s é q u e n c e of t h e existence of o n e closed géodésie in every c o n j u g a c y classe. If M is

simply connected, T h e o r e m B for géodésie flows follows from t h e foUowing resuit by G r o m o v [24]

T h e o r e m ( G r o m o v ) . Let M be a œmpact and simply connected manifold. Let g be a bumpy Riemannian metric on M. Then there exist constants a = a{g) > 0 and P = p{g) > 0 such that there are at least

0t

t

periodic geodesics of length less than t, for ail t sufficiently large.

T h e a s s u m p t i o n on t h e R i e m a n n i a n m e t r i c t o b e bumpy c o r r e s p o n d s t o o u r gener- icity a s s u m p t i o n . G e n e r a l i z a t i o n s t o géodésie flows of larger classes of R i e m a n n i a n manifolds were proved in P a t e r n a i n [34, 35] a n d P a t e r n a i n - P e t e a n [37|.

In [31], M a c a r i n i a n d Schlenk s t u d y t h e e x p o n e n t i a l g r o w t h of t h e n u m b e r of R e e b chords in spherizations. T h e y prove T h e o r e m A for R e e b c h o r d s in t e r m of t h e topology of t h e based loop space. R e s u l t s o n e x p o n e n t i a l g r o w t h r a t e of t h e n u m b e r of closed o r b i t s for certain R e e b flows on a large class of closed c o n t a c t 3 - m a n i f o l d s are proved in [10].

T h e s i m p l y c o n n e c t e d c a s e

In [3] B a l l m a n a n d Ziller improved G r o m o v ' s t h e o r e m in t h e case of simply c o n n e c t e d R i e m a n n i a n manifolds w i t h b u m p y metrics. T h e y showed t h a t t h e n u m b e r Ng{T) of closed geodesics of length less t h a n or e q u a l t o T is b o u n d e d below by t h e m a x i m u m of t h e kth b e t t i n u m b e r of t h e free loop s p a c e fc < T , u p t o s o m e c o n s t a n t d e p e n d i n g only o n t h e metric. FoUowing t h e i r idea we shall prove t h e foUowing

T h e o r e m C . Suppose that M is a compact and simply connected m-dimensional manifold. Let T, be a generic fiberwise starshaped hypersurface of T''M and R its associated Reeb vector field. Then there exist constants a = a{R) > 0 and p = PiR) > 0 such that

#OR{T) > a m a x bi{KM)

l<i<0r for ail T sufficiently large.

T w o q u e s t i o n s

I. W e a s s u m e t h e h y p e r s u r f a c e E t o b e fiberwise s t a r s h a p e d with respect to the

origin. C a n this a s s u m p t i o n b e o m i t t e d ? In t h e case of R e e b chords it c a n n o t ,

see [31].

I n t r o d u c t i o n x i II. T h e a s s u m p t i o n on E t o b e fiberwise s t a r s h a p e d is équivalent t o t h e a s s u m p -

t i o n t h a t S is of r e s t r i c t e d c o n t a c t t y p e w i t h respect t o t h e Liouville vector field Y = pdp. A r e T h e o r e m A a n d T h e o r e m B t r u e for a n y h y p e r s u r f a c e E C r * M of restricted c o n t a c t t y p e ?

T h e t h e s i s is organized as follows: In C h a p t e r 1 we i n t r o d u c e t h e définitions a n d tools t h a t we will use t h r o u g h o u t t h i s work. C h a p t e r 2 provides t h e tool of s a n d w i c h i n g u s e d h e r e t o compare t h c g r o w t h of closed R e e b o r b i t s w i t h t h e g r o w t h of closed geodesics. In C h a p t e r 3 we recall t h e définition of M o r s e - B o t t h o m o l o g y which is u s e d in t h e proof of T h e o r e m B . In C h a p t e r 4 we prove T h e o r e m A, T h e o r e m B a n d T h e o r e m C. In C h a p t e r 5 we shall e v a l u a t e o u r results on several e x a m p l e s i n t r o d u c e d in C h a p t e r 1.

In A p p e n d i x A we review s o m e t o o l s t o prove t h e c o m p a c t n e s s of m o d u l i spaces

i n t r o d u c e d in section 2.3.1. In A p p e n d i x B we recall t h e définition of t h e L e g e n d r e

t r a n s f o r m . In Appendix C we give a proof of t h e existence of G r o m o v ' s c o n s t a n t ,

see T h e o r e m 5.

Définitions and Tools

I n t h i s c h a p t e r we i n t r o d u c e t h e définitions a n d tools t h a t we will u s e t h r o u g h o u t t h i s work. In section 1.1 we d e s c r i b e t h e free loop space A M of a m a n i f o l d M

a n d i n t r o d u c e topological i n v a r i a n t m e a s u r i n g t h e topological c o m p l e x i t y of t h e free l o o p space. Section 1.2 gives a n overview of H a m i l t o n i a n d y n a m i c o n c o t a n g e n t b u n d l e s a n d fiberwise s t a r s h a p e d hypersurfaces. W e discuss t h e r e l a t i o n b e t w e e n R e e b o r b i t s o n a fiberwise s t a r s h a p e d h y p e r s u r f a c e a n d t h e 1-periodic o r b i t s of a H a m i l t o n i a n flow for which t h e h y p e r s u r f a c e is a n energy level. In section 1.3 we recall t h e définitions a n d p r o p e r t i e s of Maslov t y p e indexes i n t r o d u c e d by Conley a n d Z e h n d e r in [11] and R o b b i n a n d S a l a m o n in [39].

1.1 The free loop space

L e t {M, g) b e a connected, C ° ° - s m o o t h R i e m a n n i a n manifold. L e t A M b e t h e set of l o o p s g : 5^ —> M of Sobolev class W^''^. AM is called t h e free loop space of M.

T h i s s p a c e carries a canonical s t r u c t u r e of Hilbert manifold, see [28].

T h e e n e r g y functional £ = £g : AM —> ïïl is defined as

w h e r e = gg^t){Q{t),qit)). It i n d u c e s a filtration o n AM. For a > 0, consider t h e sublevel s e t s A" c A M of l o o p s w h o s e energy is less t h a n or e q u a l t o a,

A" := {g e A M I £{q) < a}.

T h e l e n g t h functional C := Cg : AM IR is defined by

2 1 . 1 T h e free l o o p s p a c e Similarly, for a > 0 we can consider t h e sublevel sets

£" := {qeAM\ C{q) < a}.

A p p l y i n g Schwarz's i n e q u a h t y

w i t h f { t ) = 1 a n d g{t) = \q{t)\ we see t h a t

\c\q) < £{q),

w h e r e equaUty h o l d s if a n only if q is p a r a m e t r i z e d b y arc-length.

D é n o t e by A ^ M t h e connected c o m p o n e n t of a loop a in A M . T h e c o m p o n e n t s of t h e loop s p a c e A M a r e in bijection w i t h t h e set C{M) of c o n j u g a c y classes of t h e f u n d a m e n t a l g r o u p ITI{M),

A M = ]J ^AeM.

c e C ( M )

C o u n t i n g b y c o u n t i n g c o n j u g a c y c l a s s e s i n TTI

Let X b e a p a t h - c o n n e c t e d topological space. D é n o t e b y C{X) t h e set of c o n j u g a c y classes in 7ri(X) a n d by J^iX) t h e set of free h o m o t o p y classes in AX. Given a loop a : ( 5 ' , 0 ) —>• {X,Xo) we will d é n o t e its b a s e d h o m o t o p y class in Tri{X) by [a] a n d its free h o m o t o p y class in T { X ) by | a ] .

P r o p o s i t i o n 1 . 1 . 1 . Let X be a path-connected topological space andxo a base point.

Then

$ : C{X) ^ T { X ) : [a] ^ H

is a bijection between the set of conjugacy classes in Tri{X) and the set of free ho- motopy classes in AX.

Furthermore, if f : {X,xo) —> {Y,yo) is a continuons map between based topolog- ical spaces, we have

$ o / , = / , o $ .

Proof. Let f , g : ( S \ 0 ) —> {X,Xo) b e two c o n t i n u o u s m a p s . If / is h o m o t o p i c t o g t h e n / is also freely h o m o t o p i c t o g. T h u s we get a well defined m a p

$ : 7 r i ( X ) J - ( X )

sending a b a s e d h o m o t o p y class [7] t o its free h o m o t o p y class I7]. Let 7o,7i,<a: : ( S \ 0 ) {X,xo) such t h a t

M-yo][a]-' = [71]

which is équivalent to

[a -70 • a " ^ ] = [71].

Consider t h e homotopy F : [0, l ] x [ 0 , 1 ] X defined by F ( s , t) = a ( l - ( l - s ) ( l - i ) ) . T h e n F{0, t) = a{t) and F ( l , t) = xo, meaning t h a t F is a free h o m o t o p y of curves f r o m a to XQ- Using F, one c a n construct a free h o m o t o p y of loops between q-7o-q~^

a n d 7o. As a • 70 • a^^ is b a s e d h o m o t o p i c to 71 it follows t h a t 70 is free homotopic t o 7i a n d t h u s $([70]) = ^([7i])- T h u s $ descends to a m a p

$ : C{X) ->

N o w consider a loop 7 : 5^ —> X a n d t a k e a continuons p a t h a : [0,1] X with a{0) = 7 ( 0 ) a n d q(1) = Xo- T h e n a • 7 • is a continuoxis loop w i t h base point XQ which is freely homotopic t o 7 . T h i s impUes t h a t • 7 • a~^]) = [7] which yields t h e surjectivity of

L e t [fo] a n d [fi] be two éléments of 7ri(X,a;o) with $([/o]) = ^'([/i]) a n d H : [0, l]xS^ ^ X SL free h o m o t o p y f r o m /o t o / i . Define 5 : 5^ ->• X by g{s) := H [s, 0).

T h e n g- fo- g'^ is homotopic t o f i a n d t h u s [fo] a n d [/i] are c o n j u g a t e . T h i s proves t h e injectivity of

T h e n a t u r a l i t y follows f r o m t h e définition of $ as

«'°/.([7]) = $([/°7])

= [/°7l

= / . M

= /.*([7])-

•

1.1.1 Growth coming from C{M)

Consider t h e set C{M) of c o n j u g a c y classes of t h e f u n d a m e n t a l g r o u p 7ri(M). For e a c h élément c G C(M) d é n o t e by e(c) t h e infimum of t h e energy of a closed curve r e p r e s e n t i n g c. We d é n o t e by C " ( M ) t h e set of conjugacy classes whose éléments c a n b e represented by a loop of energy a t most a,

CiM) : = {c G C ( M ) I e(c) < a} .

4 1 . 1 T h e free l o o p s p a c e T h e e x p o n e n t i a l a n d p o l y n o m i a l g r o w t h of t h e n u m b e r of c o n j u g a c y classes a s a f u n c t i o n of t h e e n e r g y a r e m e a s u r e d by

E{M) : = l i m i n f - l o g # C ° ( M ) , a n d a—>oo a

e(M) : = l i m i n f - ^ log # C ° ( M ) . a-^oo log a

1.1.2 Energy hyperbolic manifolds

Recall t h a t for a > 0, A" d é n o t e s t h e subset of loops w h o s e e n e r g y is less or equal t o a ,

A" : = {g € A M I £{q) < a}.

Let Po b e t h e set of p r i m e n u m b e r s , a n d write P : = PQ U {0}. For each p r i m e n u m b e r p d é n o t e b y Fp t h e field Z / p Z , a n d a b b r e v i a t e FQ : = <Q. T h r o u g h o u t , H, d é n o t e s singular homology. L e t

t f c : i f f c ( A " ; F p ) ^ / / f c ( A M ; F p )

b e t h e h o m o m o r p h i s m induced by t h e inclusion A ^ M ^ AM. It is well-known t h a t for each a t h e h o m o l o g y g r o u p s Hk{A'^M;¥p) vanish for ail large e n o u g h fc, see [4].

T h e r e f o r e , t h e s u m s in t h e foUowing définition a r e finite. FoUowing [20] we m a k e t h e

D é f i n i t i o n 1 . 1 . 1 . The Riemannian manifold {M, g) is energy hyperbolic if C(M,g) : = s u p l i m i n f - log d i m t ^ i / f c ( A ^ " ' ; Fp) > 0

Since M is closed, t h e p r o p e r t y energy hyperbolic d o e s n o t d é p e n d o n g while, of course, C{M,g) d o e s d é p e n d on g. W e say t h a t t h e closed m a n i f o l d M is energy hyperbolic if {M, g) is energy hyperbolic for s o m e a n d hence for a n y R i e m a n n i a n m e t r i c g o n M. For ail a > 0 we have C{M,ag) = '^C{M,g).

W e will also consider t h e polynomial growth of t h e h o m o l o g y given by c(M,g) : = s u p lim inf l o g y ^ d i m i f c H f c ( A 5 " ^ ; F p ) .

p6P "-•oo l o g n ^ F i x a R i e m a n n i a n m e t r i c g a n d p G P . It holds t h a t

d i m t o i / o ( A ^ ' ' ; F p ) = # c i ' ' ( M ) .

T h u s E{M), respectively e ( M ) , is a lower b o u n d for C{M,g), respectively c{M,g).

1.1.3 Examples

I n h i s w o r k , G r o m o v c o n j e c t u r e d t h a t " a l m o s t ail" m a n i f o l d a r e e n e r g y h y p e r b o l i c . I n d i m e n s i o n 4, P a t e r n a i n s h o w e d in [35] t h e s i m p l y c o n n e c t e d m a n i f o l d w h i c h a r e n o t e n e r g y hyperbolic u p t o h o m e o m o r p h i s m a r e

5 ^ C P ^ X 5 ^ C P ^ ^ C F ^ a n d CP^#CP^.

I n d i m e n s i o n 5, t h e simply c o n n e c t e d m a n i f o l d w h i c h a r e n o t e n e r g y h y p e r b o l i c u p t o d i f f e o m o r p h i s m axe

S\ 5 ^ X 5 ^ 52 X a n d 5 î 7 ( 3 ) / S ' 0 ( 3 ) ,

see [36]. I n his work, L a m b r e c h t s [29] s h o w e d t h a t for M i a n d M2 t w o s i m p l y c o n n e c t e d closed manifolds of t h e s a m e d i m e n s i o n a n d a field k s u c h t h a t H,{M1; k) a n d H, ( M 2 ; k) a r e not t h e c o h o m o l o g y of a s p h c r e , t h e following h o l d s

1. t h e s é q u e n c e {dimH„{A{Ml^M2); k))n>i is u n b o u n d e d ;

2. if a t least o n e of t h e c o h o m o l o g y H'f{Mi; k) is n o t a m o n o g e n i c a l g e b r a t h e n t h e s é q u e n c e {dimHn{{Ml^M2); k))n>i h a s a n e x p o n e n t i a l g r o w t h , a n d o t h e r w i s e t h i s s é q u e n c e has a l i n e a r g r o w t h .

N é g a t i v e c u r v a t u r e m a n i f o l d s

S u p p o s e o u r m a n i f o l d M c a r r i e s a R i e m a n n i a n m e t r i c of n é g a t i v e s e c t i o n a l c u r v a t u r e . P r o p o s i t i o n 1 . 1 . 2 . If M passes a Riemannian metric g of négative sectional cur-

vature, then the component of contractible loops AQM is homotopy équivalent to M, and ail other components are homotopy équivalent to .

U s i n g t h e r e s u i t of the p r e v i o u s s e c t i o n , t h i s yields C o r o U a r y 1. A M ^ M U,„„C(M) •^^

Proof. C o n s i d e r t h e e n e r g y f u n c t i o n a l £ := £g : AM —> IR w i t h r e s p e c t t o t h e m e t r i c g. I t ' s a M o r s e - B o t t f i m c t i o n a l , i.e.

crit(é:) := {g G AM I d£{q) = 0}

is a s u b m a n i f o l d of A M a n d

T , c r i t ( £ ) = k e r ( H e s s {£){q)).

M o r e o v e r i t s critical p o i n t s a r e closed geodesics. Let c b e a n o n - c o n s t a n t closed g é o d é s i e o n M. T h e n c gives rise t o a w h o l e circle of geodesics w h o s e p a r a m e t r i z a t i o n d i f ï e r b y a s h i f t t e S^. W e d é n o t e b y Se t h e set of s u c h geodesics. C o n s i d e r t h e f o l l o w i n g r e s u i t of C a r t a n [28, S e c t i o n 3.8].

6 1 . 1 T h e free l o o p s p a c e T h e o r e m 1. ( C a r t a n ) Let M be a compact manifold with strictly négative curva-

ture. Then there exists, up to parametrization, exactly one closed géodésie c in every free homotopy class which is not the class of the constant loop. c is the élément of

minimal length in its free homotopy class. AU closed geodesics on M are ofthis type.

T h u s S h a s a u n i q u e critical manifold Se in every c o m p o n e n t which is n o t t h e com- p o n e n t of t h e c o n s t a n t loops. W h i l e t h e c o m p o n e n t of t h e c o n s t a n t loop h a s as critical m a n i f o l d SQ t h e subspace of c o n s t a n t loops. Moreover ail t h e M o r s e indices axe e q u a l t o zéro. Following [23], o n e c a n résolve every critical s u b m a n i f o l d s Se into finitely m a n y n o n - d e g e n e r a t e critical p o i n t s C i , . . . , C; c o r r e s p o n d i n g t o critical p o i n t s of a M o r s e f u n c t i o n /i : Se ^ M. T h e index of a n o n - d e g e n e r a t e critical p o i n t Ci is t h e n given by t h e s u m A -I- Aj w h e r e A is t h e M o r s e index of c w i t h respect t o

€ a n d Aj t h e M o r s e index of Cj w i t h respect t o t h e M o r s e f u n c t i o n h.

Let a < 6 b e regulax values of £ a n d C i , . . . , Cfc critical p o i n t s of £ in £~^[a, b]. Let Cil,.. .Cil- b e t h e c o r r e s p o n d i n g n o n - d e g e n e r a t e critical p o i n t s of indices A j i , . . . , A^..

T h e n L e m m a 2 of [23] tells us t h a t A" is d i f ï e o m o r p h i c t o A*" w i t h a h a n d l e of index Ay a t t a c h e d for e a c h n o n - d e g e n e r a t e critical point Cij, 1 < i < k, 1 < j < ki. T h e d i f f e o m o r p h i s m c a n b e chosen t o kccp A" fixed. Using t h e m e t h o d s of Milnor in 133, Section 3], we o b t a i n t h a t t h e c o m p o n e n t of t h e c o n t r a c t i b l e loop h a s t h e h o m o t o p y t y p e of t h e s p a c e of c o n s t a n t loops while every o t h e r c o m p o n e n t h a s t h e h o m o t o p y t y p e o f S ^ •

Consider t h e c o u n t i n g f u n c t i o n CF{L) for periodic geodesics, w h e r e CF{L) = # { p e r i o d i c geodesics of l e n g t h smaller t h a n or equal t o L } .

P r o p o s i t i o n 1.1.2 tells us t h a t in t h e n é g a t i v e c u r v a t u r e case, every periodic géodésie c o r r e s p o n d t o a n élément of C{M). S e t t i n g a = \L?, we h a v e t h e following e q u a l i t y

# C ' ' ( M ) = CF{L).

A lower b o u n d for E{M) can b e d e d u c e d f r o m a resuit of Margulis.

T h e o r e m 2. (Mau-gulis 1 9 6 9 [32]) On a compact Riemannian manifold of néga- tive curvature it holds that

logCFjL) L-too L

where htop{g) is the topological entropy of the géodésie fiow. Moreover phtop{.g)L

> - 2 L -

for L large enough.

F o r a d é f i n i t i o n see [25, 5]. T h e o r e m 2 implies t h a t for L large e n o u g h ,

^C'iM) = CF{L) >

F o r e x a m p l e if M = E-^ is a n o r i e n t a b l e surface of genus 7 a n d c o n s t a n t c u r v a t u r e

— 1, t h e n htopig) = 1, see [5, Section 10.2.4.1], a n d t h u s

2L' P r o d u c t s

L e m m a 1 . 1 . 1 . Let M,N be two manifolds. Then

A ( M

AT) ^ A M

AAT.

Proof. C o n s i d e r t h e m a p </> : A ( M x N) ^ KM x AA^, sending t h e loop q : 5^ ->

M X N -.t^ {ai{t),a2{t)) o n t o ( a i , a 2 ) . •

W e will s h o w in section 5.2 t h a t t h e p r o d u c t of t w o s p h è r e s S' x S" h a s c ( M , g) > 0.

L i e g r o u p s

L e t G b e a c o m p a c t c o n n e c t e d Lie g r o u p , i.e. a c o m p a c t , c o n n e c t e d s m o o t h m a n i f o l d w i t h a g r o u p s t r u c t u r e in which t h e m u l t i p l i c a t i o n a n d inversion m a p s G x G ^ G a n d G ^ G a r e s m o o t h .

T h e f u n d a m e n t a l group 7ri(G) of a c o n n e c t e d Lie g r o u p G is a b e l i a n . In fact, c o n s i d e r i n g t h e universal cover G, t h e kernel of t h e p r o j e c t i o n p : G —> G is t h e n i s o m o r p h i c t o 7ri(G). T h i s is a discrète n o r m a l s u b g r o u p of G. Let 7 6 7ri(G).

T h e n g —>• gjg'^ is a c o n t i n u o n s m a p G —> 7ri(G). Since G is c o n n e c t e d a n d 7ri(G) d i s c r è t e , it is constant, so g'yg^^ = 7 for ail g. H e n c e 7ri(G) is c e n t r a l in G a n d in p a r t i c u l a r , it is abelian. T h i s yields

A G = ]J ^{A ^ G .}

D é n o t e b y A i M the c o m p o n e n t of t h e c o n s t a n t loop. For a > 0 we consider t h e sublevel s e t s

Al •- {q e A i M I €{q) < a}.

C h o o s e a c o n s t a n t loop 71 r e p r e s e n t i n g A i M a n d consider a n o t h e r c o m p o n e n t A j M of A G r e p r e s e n t e d by a loop 7^. Fix e G ( 0 , 1 ) . For 7 , 7 ' G A G we define

0 < i < £,

8 1 . 1 T h e free l o o p s p a c e w h e r e /i € G is s u c h t h a t /i7(0) = 7'(0). Notice t h a t ^{1) := h'y{t) is h o m o t o p i c t o 7 . T h e n t h e m a p F j : A i M —^ A j M : 7 >-> 7^ *j 7 , is a h o m o t o p y équivalence w i t h h o m o t o p y inverse A j M —> A i M : 7 i-> 7 " ^ 7. W e have

f (7 7') = -£{h^) + - ^ 5 ( 7 ' ) , for ail 7,7' e A G .

£ 1 — e

A b b r e v i a t i n g A° = A j M n A" a n d Ei = m a x { f (/17) | /i e G , 7 e A"}, we t h e r e f o r e have

F i ( A î ) c A p + ^ .

For c e 7ri(G), recall t h a t e(c) d é n o t e s t h e i n f i m u m of t h e energy of a closed c u r v e r e p r e s e n t i n g c a n d ^ ( G ) : = {c € 7ri(G) | e(c) < a } . Set £ = \- Since F^. : A i M —>

A c M is a h o m o t o p y équivalence, it follows t h a t

dimifci/fc(AÎ;Fp) < dimifc//fc(A2''+2^(^); Fp) for ail a > 0. W e c a n e s t i m a t e

d i m i i i / f c ( A ^ ; F p ) = ^ d i m t f c / f f c ( A ^ ; F p ) ceiri(G)

> ^ d i m i f c / f f c ( A f - ^ ' ^ ' ; F p )

> Y . dimtfc//fc(Aî;Fp).

ceC-(G)

W e conclude t h a t

Y,à\mLkHk{h^' j ^ ) > # C 5 " ' ( G ) • ^ d i m t f c / / f c ( A f " ' ; F p ) .

*:>0 fc>0

Considering t h e h o m o m o r p h i s m

i n d u c e d b y t h e inclusion A^ ^ A G , we c a n look a t t h e g r o w t h r a t e s Ci{G,g) : = s u p l i m i n f - l o g ^ d i m i / t i ï f c ( A f " ;Fp)

pgp n-Kx n

1 1 2

Cl (G, 5) : = s u p lim inf log d i m LkHk (Af " ; F p ) . pgp n-*oo l o g n ^ v i f /

It follows f r o m t h e définitions of E{G) a n d e ( G ) t h a t for a Lie g r o u p , CiG,g)>E{G)+C,{G,g)

a a d

c ( G , 5 ) > e ( G ) + c i ( G , 3 ) .

1.2 Cotangent bundles

L e t M b e a s m o o t h , c l o s e d m a n i f o l d of d i m e n s i o n m. L e t T*M b e t h e corre- s p o n d i n g c o t a n g e n t b u n d l e , a n d n : T*M ^ M t h e u s u a l p r o j e c t i o n . W e will d é n o t e local c o o r d i n a t e s o n M b y ç = (QI, . . . , qm), a n d o n T'M b y a; = {q,p) = {il, • • • >9m,Pl, • • • ,Pm)-

W e e n d o w e d T*M w i t h t h e standard symplœtic form u) = dX w h e r e \ = pdq =

YliLiPi dqi is t h e Liouville form. T h e d é f i n i t i o n of A d o e s n o t d é p e n d o n t h e choice of local c o o r d i n a t e s . It h a s also a global i n t e r p r é t a t i o n o n T* M a s

\{xm=p{d7rO for X e T * M a n d ^ G T^T'M.

A symplectomorphism <p : {T*M,uj) —>• {T*M,ui) is a d i f f e o m o r p h i s m s u c h t h a t t h e p u l l b a c k of t h e s y m p l e c t i c f o r m ui is ui, i.e. i^'ui = CJ.

A n Hamiltonian function i f is a s m o o t h f u n c t i o n H : T*M —> M . A n y H a m i l t o n i a n f u n c t i o n H d é t e r m i n e s a v e c t o r field, t h e Hamiltonian vector field XH d e f i n e d b y

u){X„{x),-) = -dH{x). (1.1)

L e t H : 5^ x T * M ^ IR b e a C ° ° - s m o o t h t i m e d é p e n d e n t 1 - p e r i o d i c f a m i l y of H a m i l t o n i a n functions. C o n s i d e r t h e H a m i l t o n i a n é q u a t i o n

x{t) = Xnixit)), (1.2) I n local c o o r d i n a t e s it t a k e s t h e physical f o r m

q= dpH{t,q,p) ^^^^

p = -d,H{t,q,p).

T h e s o l u t i o n s of (1.2) g e n e r a t e a f a m i l y of s y m p l e c t o m o r p h i s m s i f ' f j v i a

T h e 1 - p e r i o d i c solutions of (1.2) a r e in o n e - t o - o n e c o r r e s p o n d e n c e w i t h t h e f î x e d p o i n t s of t h e t i m e - l - m a p (pn = f n - W e d é n o t e t h e set of s u c h s o l u t i o n s b y

V{H) = {x : -^T* M \ x{t) = XH{x{t))}.

A 1 - p e r i o d i c solution of (1.2) is called non-degenerate if 1 is n o t a n eigenvalue of t h e d i f f e r e n t i a l of t h e H a m i l t o n i a n flow d(pH{x{0)) : Tc(o)T*M Tx{o)T*M, i.e.

d e t ( 7 - d^H{x{0))) + 1.

1 0 1 . 2 C o t a n g e n t b u n d l e s

If H is t i m e - i n d e p e n d e n t a n d x a n o n - c o n s t a n t 1-periodic o r b i t , t h e n it is necessaxily d e g e n e r a t e b e c a u s e x{a + t), a e JR, a r e also 1 - p e r i o d i c s o l u t i o n s . W e wilI call s u c h a n X transversally nondegenerate if t h e e i g e n s p a c e t o t h e eigenvalue 1 of t h e m a p dip]i{x{0)) : T , ( o ) T * M T^(o)T*M is o n e - d i m e n s i o n a l .

A n almost complex structure is a c o m p l e x s t r u c t u r e J o n t h e t a n g e n t b u n d l e TT'M i.e. a n a u t o m o r p h i s m J : TT*M -> TT*M s u c h t h a t = - I d . It is said t o b e u)-compatible if t h e bilinear f o r m

{•,-)=9j{;-):=w{;J-)

d e f i n e s a R i e m a n n i a n m e t r i c o n T ' M . G i v e n s u c h a n w - c o m p a t i b l e a l m o s t c o m p l e x s t r u c t u r e , t h e H a m i l t o n i a n System (1.3) b e c o m e s

XH{X) = J{x)VH{x).

T h e action functionaloî classical m e c h a n i c s AH '• A ( T * M ) - > IR a s s o c i a t e d t o H is d e f i n e d a s ^

AH{x{t)):= f {\{x{t))-H{t,x{t)))dt.

Jo

T h i s f u n c t i o n a l is C ° ° - s m o o t h a n d its critical p o i n t s a r e precisely t h e é l é m e n t s of t h e s p a c e V{H).

1.2.1 Fiberwise starshaped hypersurfaces

Let S b e a s m o o t h , c o n n e c t e d h y p e r s u r f a c e in T*M. W e say t h a t E is fiberwise starshaped if for e a c h p o i n t q £ M t h e set E , : = E n T*M is t h e s m o o t h b o u n d - a r y of a d o m a i n in T*M w h i c h is s t r i c t l y s t a r s h a p e d w i t h r e s p e c t t o t h e origin 0 , € T*M. W h e n d i m M > 2, T'M \ E h a s t w o c o m p o n e n t s . W e d é n o t e b y £>(E) t h e b o u n d e d i n n e r p a r t c o n t a i n i n g t h e zéro section a n d by D'^(E) t h e u n b o u n d e d o u t e r p a r t T*M \ L ' ( E ) , w h e r e I>(E) d é n o t e s t h e c l o s u r e of D{T.).

A h y p e r s u r f a c e E C T'M is s a i d t o b e of restricted contact type if t h e r e e x i s t s a v e c t o r field o n T * M s u c h t h a t

a n d s u c h t h a t Y is e v e r y w h e r e t r a n s v e r s e t o E p o i n t i n g o u t w a r d s . E q u i v a l e n t l y , t h e r e e x i s t s a 1 - f o r m a o n T*M s u c h t h a t da = uj a n d s u c h t h a t a A ( d a ) ' " ~ ^ is a v o l u m e f o r m o n E . O u r a s s u m p t i o n t h a t E is a f i b e r w i s e s t a r s h a p e d h y p e r s u r f a c e t h u s t r a n s l a t e s t o t h e a s s u m p t i o n t h a t E is of r e s t r i c t e d c o n t a c t t y p e w i t h r e s p e c t t o t h e Liouville vector field

m Y = Y^Pidpi

or, e q u i v a l e n t l y : t h e Liouville f o r m A defines a c o n t a c t f o r m o n E .

Figure 1.1: F i b e r w i s e s t a r s h a p e d h y p e r s u r f a c e .

T h e r e is a flow naturally associated w i t h E , g e n e r a t e d b y t h e u n i q u e v e c t o r field R a l o n g E defined by

dX{R, •) = 0, \{R) = 1.

T h e v e c t o r field R is called t h e Reeb vector field o n E , a n d its flow is called t h e Reeb flow.

For a closed o r b i t 7 of t h e R e e b flow, d é n o t e s by 7*^ its fcth i t e r a t e 7'°(t) = 7(A;t).

T h e i t é r â t e s of a closed o r b i t s have t h e s a m e i m a g e a n d a r e t h u s geometrically t h e s a m e . W e a r e interested in t h e g r o w t h of geometrically distinct closed orbits. W e t h u s i n t r o d u c e t h e set OR of simple closed orbits of t h e R e e b flow c o n t a i n i n g ail t h e R e e b o r b i t s which aie geometrically différent.

For T > 0 let OR^T) b e t h e set of s i m p l e closed o r b i t s of tp^ w i t h p e r i o d < r . W e

m e a s u r e t h e g r o w t h of t h e n u m b e r of é l é m e n t s in OII{T) by

1 2 1 . 2 C o t a n g e n t b u n d l e s

NR : = l i m i n f - l o g ( # C » f l ( T ) ) , T—>oo T

nR : = l i m i n f - ^ l o g ( # O f l ( r ) ) . T-K» log T

T h e n u m b e r NR is t h e exponential growth rate of closed o r b i t s , while UR is t h e polynomial growth rate.

1.2.2 Dynamics on fiberwise starshaped hypersurfaces

G i v e n a fiberwise s t a r s h a p e d h y p e r s u r f a c e S c T*M, we c a n define a n H a m i l t o n i a n f u n c t i o n F : r * M - > B, b y t h e t w o c o n d i t i o n s

F I E S I , F{q,sp) = s^F{q,p), s > 0 and {q,p) £ T'M. (1.4)

T h i s f u n c t i o n is of class C^, fiberwise h o m o g e n e o u s of d e g r e e 2 a n d s m o o t h ofF t h e zero-section.

L e m m a 1 . 2 . 1 . Let E C T*M be a fiberwise starshaped hypersurface. If Y, is the level set of a Hamiltonian function H : T'M IR, then the Reeb flow of X is a reparametrization of the Hamiltonian flow.

Proof. T h e r e s t r i c t i o n of t h e 2 - f o r m w = dX t o T S i s d e g e n e r a t e a n d o f r a n k ( 2 m — 2 ) . I t s kernel is t h e r e f o r e l - d i m e n s i o n a l . B y d é f i n i t i o n of b o t h R e e b a n d t h e H a m i l t o n i a n v e c t o r fields, t h e y also d e f i n e t h i s kernel since

iflrfAlrK = 0 a n d

tA-„w|n: = -dHln: = 0.

T h e r e f o r e

^H{X) = a{x)R{x)

for e v e r y a; € S , w i t h a n o w h e r e v a n i s h i n g s m o o t h f u n c t i o n a. •

C o n d i t i o n (1.4) t h u s implies t h a t t h e H a m i l t o n i a n v e c t o r field Xp r e s t r i c t e d t o E is a positive r e p a r a m e t r i z a t i o n of t h e R e e b vector field. C o n s e q u e n t l y

for every a; 6 E a n d for a s m o o t h p o s i t i v e f u n c t i o n cr o n IR x E . In p a r t i c u l a r , we h a v e t h a t

w h e r e s is s m o o t h and positive. Since E is c o m p a c t , s is b o u n d e d from a b o v e , s a y s{x) < b. T h u s if 7 is a 1 - p e r i o d i c o r b i t of X p , t h e n 7 is a p e r i o d i c o r b i t of Xn w h o s e p e r i o d is a t most b. H e n c e a <-periodic o r b i t of Xp is a p e r i o d i c o r b i t of Xn w h o s e p e r i o d is less t h a n {t + l)b, w h i c h îOT t > 1 is less t h a n 2bt. T h u s , if w e d é n o t e b y O f ( * ) t h e set of closed o r b i t s oi tfip w i t h p e r i o d < t, we h a v e t h a t

#Onit) >

T h e g r o w t h of t h e f u n c t i o n 11-^ ij^Opit) is t h u s e q u a l t o t h e g r o w t h o î t ^ jfOjî{t)

N o w we w a n t t o establish a c o r r e s p o n d e n c e b e t w e e n 1-periodic s o l u t i o n s of XH a n d closed o r b i t s of XH on E . C o n s i d e r t h e r a d i a l m a p c^, : T * M - > T*M,

Cs{x) := SX := {q,sp) t h e n t h e v e c t o r field V o n T * M d e f î n e d b y

Y{x) •- ^ Cs{x).

as s = i

is t h e Liouville vector field, in local c o o r d i n a t e s Y{q,p) = ^iPidpi- In f a t c , difîer- e n t i a t i n g c*A = J2i^Pi'^1i ~ w i t h r e s p e c t t o s, we o b t a i n

A = CyX = iyi^-

D i f f e r e n t i a t i n g F{sx) = s^F{x) w i t h r e s p e c t t o s a t s = 1 we get E u l e r ' s i d e n t i t y 2F{x) = dF{x){Y{x)) = -UJ{XF{X),Y{X)) = \ { X F { X ) ) .

I n v i e w of c * F = s^F a n d c* w = s w, we g e t

c*i((c.).XF)W = i x F ( c »

= s ixpOj

= -sdF

= - ^ d ( c : F )

1

T h u s {cs)tXF = -Xp or e q u i v a l e n t l y dCs{x){XF){x) = ^XF{SX).

F i x a 1 - p e r i o d i c solution x of Xp a n d d e f i n e Xg b y Xs{t) := sx{t). T h e n

Xs = dCsix){Xp){x) = -XF{XS).

1 4 1 . 2 C o t a n g e n t b u n d l e s

S u p p o s e Apix) = a > 0. E u l e r ' s i d e n t i t y yields

AF{X) = f \ { X F { X ) ) - F{x) = F{x).

Jo

So if we set s := l/y/â, t h e n satisfies F{Xs) = s^F{x) = 1. T h u s Xg is a closed orbit of p e r i o d ^/â.

Conversely if y : 5^ —> E is a closed c h a r a c t e r i s t i c , t h e n y = ^Xpiy) for s o m e s > 0.

T h u s x{t) := ly{t) is a l - p e r i o d i c s o l u t i o n of Xp a n d x^ = y.

We o b t a i n t h a t t h e m a p x ^ i / ^ is a b i j e c t i o n b e t w e e n l - p e r i o d i c s o l u t i o n s of Xp on T*M w i t h a c t i o n a a n d t h e closed o r b i t s of Xj{ o n E w i t h p e r i o d y/â. C o u n t i n g closed o r b i t s of X[{ o n E w i t h p e r i o d less t h a n or equal t o a is t h u s é q u i v a l e n t t o c o u t i n g l - p e r i o d i c s o l u t i o n s of Xp o n T*M w i t h a c t i o n less t h a n or e q u a l t o a^.

In t h e sequel of t h i s work, it will b e e v e n m o r e convenient t o consider l - p e r i o d i c s o l u t i o n s of XaF on T'M w i t h a c t i o n less t h a n or equal t o a.

1.2.3 Spherization of a cotangent bundle

We recall f r o m [31]. Let M b e a closed c o n n e c t e d m a n i f o l d a n d E C T'M a fiberwise s t a r s h a p e d h y p e r s u r f a c e in T'M. T h e h y p e r p l a n e field

e 5 : : = k e r ( A | s ) C T E

is a c o n t a c t s t r u c t u r e on E . Consider a n o t h e r fiberwise s t a r s h a p e d h y p e r s u r f a c e E ' . T h e r a d i a l p r o j e c t i o n in e a c h fiber i n d u c e s a m a p i/;, : E , ^ R s u c h t h a t for e v e r y p £ E ç , il>q{p)p G E^. T h e n t h e difîerential of t h e d i f f e o m o r p h i s m

satisfies

* Ê E ' ( A | E ) = V ' A | E

w h e r e ip{q,p) = •>pq{p)- T h u s Î'EE' m a p s t o ^ n d hence is a c o n t a c t o m o r p h i s m ( E , ^ 5 : ) ^ ( S ' , e É ) -

T h e i n d u c e d équivalent class of t h o s e c o n t a c t m a n i f o l d is called t h e spherization (SM,^) of t h e c o t a n g e n t b u n d l e . T h e u n i t c o s p h e r e b u n d l e {SiM{g),keT X),

SiM{g) : = {{q,p) 6 T'M \ \p\ = 1}

associated t o a R i e m a n n i a n m e t r i c 5 o n M is a p a r t i c u l a r r e p r é s e n t a t i v e .

F i x a r e p r é s e n t a t i v e ( E , Ç s ) . For e v e r y s m o o t h p o s i t i v e f u n c t i o n / : E —> M , | s = k e r ( / A | 5 : ) h o l d s t r u e . W e c a n t h u s c o n s i d e r t h e a s s o c i a t e d R e e b v e c t o r field Rf o n T E d e f i n e d a s t h e u n i q u e v e c t o r field s u c h t h a t

d { f \ ) { R f , •) = (), f X i R f ) = l.

W e h a v e s e e n i n t h e p r e v i o u s section t h a t for / = 1 a n y H a m i l t o n i a n f u n c t i o n H : T*M —>• IR w i t h H~^{1) = E , w h e r e 1 is a r e g u l a r v a l u e , t h e R e e b flow of Rf is a t i m e c h a n g e of t h e H a m i l t o n i a n flow ip\j r e s t r i c t e d t o E . N o t e t h a t for a d i f f é r e n t f u n c t i o n / t h e R e e b flows o n E c a n b e c o m p l e t e l y d i f f é r e n t .

G i v e n a n o t h e r r e p r é s e n t a t i v e E ' , t h e a s s o c i a t e d c o n t a c t o m o r p h i s m Î'SE' c o n j u g a t e s t h e R e e b flows o n ( E , A ) a n d (E',V'~^A), in f a c t

d^MRx) = R^-ix-

T h i s j a e l d s t h a t t h e set of R e e b flows o n {SM, ^ ) is i n b i j e c t i o n w i t h H a m i l t o n i a n flows o n fiberwise s t a r s h a p e d h y p e r s u r f a c e s , u p t o t i m e c h a n g e .

T h e o r e m A a n d T h e o r e m B t h u s give lower b o u n d s for t h e g r o w t h r a t e of closed o r b i t s for any Reeb flow on the spherization S M o f T ' M .

1.3 Maslov index

C o n s i d e r IR^'" e n d o w e d w i t h its s t a n d a r d s y m p l e c t i c s t r u c t u r e ujo = dpA dq, {q,p) e WT x H ™ ,

a n d i t s s t a n d a r d c o m p l e x s t r u c t u r e

D é n o t e b y 5 p ( 2 m ) t h e s e t of s y m p l e c t i c a u t o m o r p h i s m s of ( H ^ ^ j W o ) , i.e.

Sp{2m) := { * € M a t ( 2 m x 2 m , IR) | ¥ J o ^ = J o } , b y £{m) t h e s p a c e of L a g r a n g i a n s u b s p a c e s of (K^™, WQ), i.e.

£ ( m ) : = {L c K^™ | uio{v,w) = 0 for ail v,w e L a n d d i m L = m}, a n d b y AQ t h e vertical L a g r a n g i a n s u b s p a c e AQ = { 0 } x H " .

T h e M a s l o v i n d e x a s s o c i â t e s a n integer fi{'if) t o e v e r y l o o p '9 : IR/Z —> Sp{2m) of s y m p l e c t i c m a t r i c e s . It s a t i f i e s a n h o m o t o p y a x i o m , t w o l o o p s a r e h o m o t o p i c if a n d o n l y if t h e y h a v e t h e s a m e M a s l o v index. In t h e following w e i n t r o d u c e t w o M a s l o v t y p e i n d e x e s t h a t we will u s e in t h e g r a d i n g of F l o e r h o m o l o g y .

1 6 1 . 3 M a s l o v i n d e x

1.3.1 Maslov index for symplectic path

I n [11] C o n l e y a n d Z e h n d e r i n t r o d u c e d a M a s l o v t y p e i n d e x t h a t a s s o c i â t e s a n integer M c z ( ^ ) t o every p a t h of s y m p l e c t i c a u t o m o r p h i s m s b e l o n g i n g t o t h e s p a c e

SP := : [0,r] ^ Sp{2m) \ ^{0) = / , d e t ( / - ^ ( T ) ) ^ 0}.

T h e foUowing d e s c r i p t i o n is p r e s e n t e d in [41].

It is well k n o w n t h a t t h e q u o t i e n t Sp{2m)/U{m) is c o n t r a c t i b l e a n d so t h e f u n d a - m e n t a l g r o u p of Sp{2m) is i s o m o r p h i c t o Z . T h i s i s o m o r p h i s m c a n b e r e p r e s e n t e d b y a n a t u r a l c o n t i n u o u s m a p

p : Sp{2m)

w h i c h r e s t r i c t s t o t h e d é t e r m i n a n t m a p o n Sp{2m) Ci 0{2m) ~ U{m). C o n s i d c r t h e set

Sp{2my := G Sp{2m) \ d e t ( / - «') ^ 0}.

It h o l d s t h a t Sp{2Tn)* h a s t w o c o n n e c t e d c o m p o n e n t s 5 p ( 2 m ) ± : = {«f € Spi2m) \ ± d e t ( / - > 0}.

M o r e o v e r , every l o o p in 5 p ( 2 m ) ' is c o n t r a c t i b l e in Sp{2Tn).

For a n y p a t h ^' : [0, r] —> Sp{2m) choose a f u n c t i o n a : [0, r] —> IR s u c h t h a t

= e ' " ( " a n d d e f i n e

TT

For A € Sp{2my choose a p a t h ^ ^ ( i ) € Sp{2mY such t h a t * ^ ( 0 ) = A a n d * > i ( l ) S

{ - / , d i a g ( 2 , - 1 , . . . , - 1 , i , - 1 , . . . , - 1 ) } . T h e n A i ( ^ ' ^ ) is i n d e p e n d e n t of t h e choice of t h i s p a t h . D e f i n e

r{A) = ù.,{^A), AeSp{2my.

T h e Conley-Zenhder index of a p a t h ^ e <SF is define a s t h e i n t e g e r

T h e foUowing i n d e x i t é r a t i o n f o r m u l a , p r o v e d in [41], will b e u s e d in s e c t i o n 4.2.1.

L e m m a 1 . 3 . 1 . Let ^ ( t ) € Sp{2m) be any path such that

^{kT + t) = I ' ( t) ^ ' ( r ) * fort>0 and k e N . Then

A f c . ( f ) = fcAi(*) for every k e N . Moreover, \r{A)\ < n for every A e Sp{2my.