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Comptes Rendus Mathématique
Volume 355, n° 8
pages 914-919 (août 2017)
Doi : 10.1016/j.crma.2017.07.010
Received : 13 June 2017 ;  accepted : 24 July 2017
A notion of Denjoy sub-system
Une notion de sous-système de Denjoy
 

Marie-Claude Arnaud a, b , Patrice Le Calvez c
a Avignon Université, Laboratoire de mathématiques d'Avignon (EA 2151), 84018 Avignon, France 
b Institut universitaire de France, France 
c Sorbonne Universités, UPMC Université Paris-6, Institut de mathématiques de Jussieu—Paris Rive Gauche, UMR 7586, CNRS, Université Paris-Diderot, Sorbonne Paris Cité, 75005 Paris, France 

Abstract

We introduce a notion of Denjoy sub-system that generalizes that of the Aubry–Mather set. For such systems, we prove a result similar to Denjoy theorem (non-existence of   Denjoy sub-systems), and study their Lyapunov exponents.

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Résumé

Nous introduisons une notion de sous-système de Denjoy qui généralise celle d'ensemble d'Aubry–Mather. Pour ces systèmes, nous montrons un analogue du théorème de Denjoy (la non-existence de sous-systèmes de Denjoy de classe  ) et étudions leurs exposants de Lyapunov.

The full text of this article is available in PDF format.
Version française abrégée

Nous introduisons une notion de sous-système de Denjoy pour un difféomorphisme d'une variété, notion qui généralise celle d'ensemble d'Aubry–Mather.

Définition 0.1

Soit   un difféomorphisme de classe   d'une variété M . Un   (resp. Lipschitz) sous-système de Denjoy pour f est un triplet   où

  est un plongement de classe   (resp. biLipschitz) ;
  est un contre-exemple de Denjoy d'ensemble minimal   ;
  ;
 .

Remarque 1

(i)
Pour les difféomorphismes exact symplectiques de l'anneau qui dévient la verticale, la théorie d'Aubry–Mather fournit de nombreux sous-systèmes de Denjoy (voir [[7]]).
(ii)
D'autres ensembles de Cantor qui ne sont pas des sous-systèmes de Denjoy apparaissent de manière naturelle en dynamique. Par exemple, un fer à cheval a des points périodiques et un odomètre a des sous-ensembles fermés non triviaux périodiques.

Notre premier théorème énonce qu'une dynamique de Denjoy ne peut pas être insérée dans une courbe (non nécessairement invariante) de classe  .

Théorème 0.2

Il n'existe pas de sous-système de Denjoy de classe  .

Corollaire 0.3

Soit   un difféomorphisme de classe   qui dévie la verticale (et n'est pas forcément symplectique). Si   est un ensemble d'Aubry–Mather de nombre de rotation irrationnel contenu dans le graphe d'une fonction de classe    , alors E coïncide avec ce graphe.

L'entropie topologique et métrique d'un sous-système de Denjoy est nulle. Le résultat suivant exprime que l'unique mesure de probabilité définie par un sous-système de Denjoy de classe   a au moins un exposant de Lyapunov nul.

Théorème 0.4

Soit f un difféomorphisme de classe   d'une variété M. Si   est un sous-système de Denjoy de classe   pour f, alors l'unique mesure de probabilité borélienne f invariante à support dans   a au moins un exposant de Lyapunov nul.

Corollaire 0.5

Soit f un difféomorphisme de classe   qui préserve l'aire d'une surface M. Si   est un sous-système de Denjoy de classe   pour f, alors l'unique mesure de probabilité borélienne f invariante à suppport dans   a ses deux exposants de Lyapunov nuls.

Remarque 2

(i)
Le second auteur a montré dans [[6]] qu'un difféomorphisme générique symplectique déviant la verticale a beaucoup d'ensembles d'Aubry–Mather qui sont uniformément hyperboliques. Parmi ceux-ci, ceux qui ont un nombre de rotation irrationnel définissent alors des sous-systèmes de Denjoy qui sont Lipschitz, mais ne sont pas de classe  .
(ii)
Dans [[2]], il est montré que l'unique mesure de probabilité invariante portée par un lacet invariant d'un difféomorphisme de classe   ne contenant pas de point périodique a au moins un exposant de Lyapunov nul. La première remarque montre que ce résultat ne peut pas être étendu aux sous-systèmes de Denjoy Lipschitz définis à l'aide d'une courbe non invariante.
(iii)
Le premier auteur a montré dans [[1]] une sorte de réciproque au Corollaire 0.5 : si les exposants de Lyapunov de la mesure à support dans un ensemble d'Aubry–Mather de nombre de rotation irrationnel d'une application symplectique déviant la verticale sont nuls, alors le support de la mesure est en un certain sens de classe   presque partout. La proposition suivante montre que ce résultat est faux si on enlève l'hypothèse de déviation de la verticale.

Proposition 0.6

Il existe un difféomorphisme de classe   F de   qui admet un sous-système Lipschitz de Denjoy   tel que

les exposants de Lyapunov de l'unique mesure de probabilité à support dans   sont nuls ;
il n'existe pas de sous-système de Denjoy de classe     pour un difféomorphisme de classe   f de   tel que  .

Remarque 3

L'exemple de sous-système de Denjoy construit dans la Proposition 0.6 admet en fait un feuilletage Lipschitz de courbes invariantes dont chacune porte un contre-exemple de Denjoy.

Cette notion de sous-système de Denjoy s'étend pour les flots.

Définition 0.7

Soit   un flot de classe   d'une variété M . Un sous-flot de Denjoy de classe   (resp. Lipschitz) pour   est un triplet   où

  est un plongement de classe   (resp. Lipschitz) ;
  est un flot de classe   sur   sans point périodique ni orbite dense ;
si K est l'unique ensemble fermé minimal invariant de  , alors   est invariant par   ;
pour tout   et tout  , on a  .

Théorème 0.8

Il n'existe pas de sous-flot de Denjoy de classe  .

Un résultat similaire pour les graphes est prouvé dans [[4]] dans le cadre de la théorie K.A.M. faible.

Remerciements. Les auteurs remercient Raphaël Krikorian de leur avoir posé la question de la régularité   des ensembles d'Aubry–Mather.

Introduction

In 1932, Arnaud Denjoy proved in [[3]] that if a   vector-field of   has a rotation vector that does not belong to  , then every orbit is dense in  . A similar result for diffeomorphisms of   can be stated in the following way: if an orientation-preserving   diffeomorphism of the circle has an irrational rotation number, then it is   conjugated to a rotation. There exist examples of orientation-preserving   diffeomorphisms of   that have an irrational rotation number and are not   conjugated with a rotation, or equivalently are not minimal (see [[5]] for example). Such examples in any regularity (i.e.   or  ) are now called Denjoy counter-examples , and they have a unique minimal invariant closed set that is a Cantor set (i.e. compact, totally disconnected and with no isolated point).

This implies of course that if a   diffeomorphism f of a manifold has an invariant loop Γ such that   has an irrational rotation number and is not minimal, then Γ is not  . Here we address the question when we do not require the curve to be invariant: can a Denjoy dynamics be contained in some non-invariant   curve? In other words, can the set that supports this dynamics be   in some sense? To explain in detail this question, we have to introduce some notions.

Definition 1.1

Let   be a   diffeomorphism of a manifold M . A   (resp. Lipschitz) Denjoy sub-system for f is a triplet   where

  is a   (resp. biLipschitz) embedding;
  is a Denjoy counter-example with invariant minimal set  ;
 ;
 .

Remark 1

(i)
For area-preserving twist maps of the annulus, Aubry–Mather theory tells us that there are many Lipschitz Denjoy sub-systems (see [[7]]).
(ii)
Other kinds of invariant Cantor sets appear naturally in dynamical systems that are not Denjoy sub-systems. For example, a horseshoe has periodic points, an odometer has periodic non-trivial closed subsets.

Our first result asserts that a Denjoy sub-system is never  .

Theorem 1.2

There exist no   Denjoy sub-system.

We immediately get:

Corollary 1.3

Let   be a   twist map (non necessarily area preserving). If   is an Aubry–Mather set of irrational rotation number that is contained in the graph of a   function  , then E coincides with this graph.

The topological entropy of a circle homeomorphism is equal to zero. Moreover, such a homeomorphism is uniquely ergodic in case its rotation number is irrational. Consequently the restriction of a diffeomorphism of a manifold to a Denjoy sub-system has zero entropy and is uniquely ergodic. The next result asserts that the unique invariant probability measure has one zero Lyapunov in case the sub-system is  .

Theorem 1.4

Let f be a   diffeomorphism of a manifold M. If   is a   Denjoy sub-system for f, then the unique f invariant Borel probability measure that is supported on   has at least one vanishing Lyapunov exponent.

We easily deduce the following.

Corollary 1.5

Let f be an area-preserving   diffeomorphism of a surface M. If   is a   Denjoy sub-system for f, then the unique f invariant Borel probability measure supported on   has its two Lyapunov exponents equal to zero.

Remark 2

(i)
The second author proved in [[6]] that, for a generic area-preserving twist map, many Aubry–Mather sets are uniformly hyperbolic. Such an Aubry–Mather set cannot be a loop. Among them, those that have an irrational rotation number are Lipschitz Denjoy sub-systems, but not   Denjoy sub-systems.
(ii)
It was proved in [[2]] that any continuous invariant loop by a   diffeomorphism of a surface that contains no periodic point carries a unique invariant measure that has at least one zero Lyapunov exponent. The first remark shows that this result cannot be extended to Lipschitz Denjoy sub-systems that are not defined via an invariant curve.
(iii)
The first author proved in [[1]] a kind of reverse result of Corollary 1.5: if the Lyapunov exponents of a measure that is supported on an Aubry–Mather set with irrational rotation number of a symplectic twist map are zero, then the support of the measure is, in some sense,   regular almost everywhere. The next proposition gives a counter-example in the non-twist setting.

Proposition 1.6

There exists a   diffeomorphism F of   that admits a Lipschitz Denjoy sub-system   such that

the Lyapunov exponents of the unique invariant probability measure that is supported on   are equal to zero;
there exists no   Denjoy sub-system   for a   diffeomorphism f of   such that  .

Remark 3

The diffeomorphism built in Proposition 1.6 has a Lipschitz foliation into invariant Lipschitz graphs on which the dynamics is a Denjoy counter-example.

This notion of Denjoy sub-system can be extended to flows.

Definition 1.7

Let   be a   flow on a manifold M . A   (resp. Lipschitz) Denjoy sub-flow for   is a triplet   where

  is a   (resp. Lipschitz) embedding;
  is a   flow of   with neither periodic orbit nor dense orbit;
if K is the unique minimal invariant closed subset of  , then   is invariant by  ;
for every   and every  , one has  .

Theorem 1.8

There exists no   Denjoy sub-flow.

In [[4]], a similar result for graphs is proved in the setting of weak K.A.M. theory.

The key result, in the proofs of Theorem 1.2, Theorem 1.4 is the following:

Proposition 2.1

Let   be some integer. Let   be a   Denjoy sub-system for some  -diffeomorphism  . There exists an orientation-preserving   diffeomorphism   such that  .

Proof

With the notations of Proposition 2.1, observe that the two loops   and   are tangent along the f invariant Cantor set  . Indeed, this set is included in both loops and has no isolated point.

Let   be a tubular neighborhood of   in which we can define a   projection  . Observe that  , coinciding with   on K , is a   embedding when restricted to a neighborhood of K . This implies that there exist two open neighborhoods U ,   of K in   such that   and such that   induces a   diffeomorphism between U and  . The family of connected components of U is an open covering of K . By compactness of K , there are finitely many components that meet K . So, taking a subset of U if necessary, we can assume that the number of connected components of U is finite, larger than 1, and that each connected component meets K . We can index these components, getting a family   ordered in the usual cyclic order along  . Setting  , one gets another family   ordered in the usual cyclic order along   because  . We can find a covering   of K by open and connected subsets of  , satisfying   for every  . For every  , we define  , and we denote by   the connected component of the complement of   that lies between   and   and   the connected component of the complement of   that lies between   and  . To extend   to a   diffeomorphism of  , it is sufficient to find, for every  , an increasing   diffeomorphism   with prescribed values of the derivatives  ,  , at the two ends of   to ensure a global differentiability. There is no obstruction to such a construction.  □

Denjoy's Theorem implies that under the assumption of Proposition 2.1, the integer k must be equal to 1, so Theorem 1.2 is proved. Moreover, when  , then   is a Df invariant lines bundle along   that satisfies  . So, for every integer  , one has
Dfn⋅γ′=(φn)′(γ′∘φn). We deduce that the unique f invariant probability measure supported on   has a Lyapunov exponent that coincides with the Lyapounov exponent of the unique invariant measure of φ , a   diffeomorphism of  . This Lyapunov exponent being known to vanish, Theorem 1.4 is proved.

Proof of Proposition 1.6

As explained in [[5]], one can construct a   Denjoy counter-example   such that:

g has exactly two orbits of wandering intervals   and   where  ;
 ;
 ;
the derivative   is equal to 1 on the minimal compact invariant set K .

In particular, one has  .

There exists a Lipschitz function  , uniquely defined up to an additive constant, such that
ψ|⋃k∈ZInt(Ik)′=1,ψ|⋃k∈ZInt(Jk)′=−1. Observe that   is constant along every orbit of g that is contained in  .

The map   defined by   is also Lipschitz and differentiable on  . Moreover, for every  , one has
η′(θ)=g′(θ)ψ′(g(θ))−ψ′(θ)=ψ′(θ)(g′(θ)−1). The fact that   is bounded on   and that   is continuous on   and vanishes on K implies that η is  .

Finally, let us define a   diffeomorphism F of   by the formula
F(θ,r)=(g(θ),r+η(θ)). The graph of ψ is invariant by F becauseF(θ,ψ(θ))=(g(θ),ψ(θ)+η(θ))=(g(θ),ψ∘g(θ)).

Note that for every   and every  , one has
DFk(θ0,ψ(θ0))=((gk)′(θ0)0bk1). Hence the two Lyapunov exponents of the unique invariant measure that is supported on the graph of ψ are zero.

Moreover, if  , there exists two sequences   and   that converge to θ . The slope of ψ between the two ends of   is 1 and the slope of ψ between the two ends of   is −1. This implies that there exists no   Denjoy sub-system   of a diffeomorphism f of   such that  .

Proof of Theorem 1.8

We assume that   is a   Denjoy sub-flow for some   flow   on a manifold M . We choose on   an essential   embedding   that is transverse to the flow direction   of  . The first return map   is then well defined, of class  , and the map   is a   Denjoy counter-example. We denote K the minimal set of h and set   and  .

Let   be a tubular neighborhood of   in which we can define a   projection   and consider the   hypersurface  . The flow direction   of   is transverse to   on  . Moreover, the first return map of   on   is well defined on   and coincide with  . One deduces that there exist two neighborhoods   and   of   in   such that the first return map of   on   is well defined on   and induces a   diffeomorphism   between   and  . The map   is a   map defined on a neighborhood of K with values in   that coincides with the   diffeomorphism h on K . So it induces a   diffeomorphism between two neighborhoods U ,   of K . As in the proof of Proposition 2.1, we may require that U and   have finitely many connected components and we can construct a   diffeomorphims of   that coincides with h in a neighborhood of K . This gives a contradiction.


Acknowledgement

Marie-Claude Arnaud is supported by the “Institut universitaire de France”.

References

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Le Calvez P. Les ensembles d'Aubry–Mather d'un difféomorphisme conservatif de l'anneau déviant la verticale sont en général hyperboliques The Aubry–Mather sets of a conservative diffeomorphism of the annulus twisting the vertical are hyperbolic in general C. R. Acad. Sci. Paris, Ser. I 1988 ;  306 (1) : 51-54(in French).
Mather J.N. Existence of quasiperiodic orbits for twist homeomorphisms of the annulus Topology 1982 ;  21 (4) : 457-467 [cross-ref]



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