Nonuniform hyperbolicity for C1-generic diffeomorphisms

We study the ergodic theory of non-conservative C ¹-generic diffeomorphisms. First, we show that homoclinic classes of arbitrary diffeomorphisms exhibit ergodic measures whose supports coincide with the homoclinic class. Second, we show that generic (for the weak topology) ergodic measures of C ¹-generic diffeomorphisms are non-uniformly hyperbolic: they exhibit no zero Lyapunov exponents. Third, we extend a theorem by Sigmund on hyperbolic basic sets: every isolated transitive set Λ of any C ¹-generic diffeomorphism f exhibits many ergodic hyperbolic measures whose supports coincide with the whole set Λ.     

Periodic orbits and chain-transitive sets of C1-diffeomorphisms

We prove that the chain-transitive sets of C¹-generic diffeomorphisms are approximated in the Hausdorff topology by periodic orbits. This implies that the homoclinic classes are dense among the chain-recurrence classes. This result is a consequence of a global connecting lemma, which allows to build by a C¹-perturbation an orbit connecting several prescribed points. One deduces a weak shadowing property satisfied by C¹-generic diffeomorphisms: any pseudo-orbit is approximated in the Hausdorff topology by a finite segment of a genuine orbit. As a consequence, we obtain a criterion for proving the tolerance stability conjecture in Diff¹(M).     

Récurrence et généricité

Résumé. Nous montrons un lemme de connexion C¹ pour les pseudo-orbites des difféomorphismes des variétés compactes. Nous explorons alors les conséquences pour les difféomorphismes C¹-génériques. Par exemple, les difféomorphismes conservatifs C¹-génériques (dune variété connexe) sont transitifs.

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We prove a C¹-connecting lemma for pseudo-orbits of diffeomorphisms on compact manifolds. We explore some consequences for C¹-generic diffeomorphisms. For instance, C¹-generic conservative diffeomorphisms (on connected manifolds) are transitive.

     

Codimension one generic homoclinic classes with interior

We study C ¹-generic diffeomorphisms with a homoclinic class with non empty interior and in particular those admitting a codimension one dominated splitting. We prove that if in the finest dominated splitting the extreme subbundles are one dimensional then the diffeomorphism is partially hyperbolic and from this we deduce that the diffeomorphism is transitive.     

The set of axiom A diffeomorphisms with no cycles

LetM be aC ^∞ closed manifold and Diff¹ (M) be the space of diffeomorphisms ofM endowed with theC ¹ topology. This paper contains an affirmative answer to the following conjecture raised by Mañé, which is an extension of the stability and Ω-stability conjectures of Palis and Smale, as follows: theC ¹ interior of the subset of diffeomorphism such that all the periodic points are hyperbolic is characterized as the set of diffeomorphisms satisfying Axiom A and the no-cycles condition. Moreover, it is showed that theC ¹ interior of the set of all Kupka-Smale diffeomorphisms coincides with the set of all diffeomorphisms satisfying Axiom A and the strong transversality condition.     

Generic robustness of spectral decompositions

We prove that given a compact n-dimensional boundaryless manifold M, n?2, there exists a residual subset R of Diff¹(M) such that if f∈R admits a spectral decomposition (i.e., the non-wandering set admits a partition into a finite number of transitive compact sets), then this spectral decomposition is robust in a generic sense (tame behavior). This implies a C¹-generic trichotomy that generalizes some aspects of a two-dimensional theorem of Mañé [Topology 17 (1978) 386-396].Lastly, Palis [Astérisque 261 (2000) 335-347] has conjectured that densely in Diff^k(M) diffeomorphisms either are hyperbolic or exhibit homoclinic bifurcations. We use the aforementioned results to prove this conjecture in a large open region of Diff¹(M).     

Dynamics of generic 2-dimensional linear differential systems

We prove that for a C⁰-generic (a dense Gδ) subset of all the 2-dimensional conservative nonautonomous linear differential systems, either Lyapunov exponents are zero or there is a dominated splitting μ almost every point.     

New criteria for hyperbolicity based on periodic sets

We prove some criteria for uniform hyperbolicity based on the periodic points of the transformation. More precisely, if a mild hyperbolicity condition holds for the periodic points of any diffeomorphism in a residual subset of a C ¹-open set U then there exists an open and dense subset A ? U of Axiom A diffeomorphisms. Moreover, we also prove a noninvertible version of Ergodic Closing Lemma which we use to prove a counterpart of this result for local diffeomorphisms.     

Actions de groupes de Kazhdan sur le cercle

We prove that every homomorphism from a discrete Kazhdan group to the group of orientation-preserving diffeomorphisms of the circle of class C^1+α (α>1/2) has a finite image.     

Interiors of sets of vector fields with shadowing corresponding to certain classes of reparameterizations

The structure of the C ¹-interiors of sets of vector fields with various forms of the shadowing property is studied. The fundamental difference between the problem under consideration and its counterpart for discrete dynamical systems generated by diffeomorphisms is the reparameterization of shadowing orbits. Depending on the type of reparameterization, Lipschitz and oriented shadowing properties are distinguished. As is known, structurally stable vector fields have the Lipschitz shadowing property. Let X be a vector field, and let p and q be its points of rest or closed orbits. Suppose that the stable manifold of p and the unstable manifold of q have a nontransversal intersection point. It is shown that, in this case, the vector field X does not have the Lipschitz shadowing property. If one of the orbits p and q is closed, then X does not have the oriented shadowing property. These assertions imply that the C ¹-interior of the set of vector fields with the Lipschitz shadowing property coincides with the set of structurally stable vector fields. If the dimension of the manifold under consideration is at most 3, then a similar result is valid for the oriented shadowing property. We study the structure of the C ¹-interiors of sets of vector fields with various forms of the shadowing property. It is shown that, in the case of the Lipschitz shadowing property, it coincides with the set of structurally stable systems. For manifolds of dimension at most 3, a similar result is valid for the oriented shadowing property.     

Smooth diffeomorphisms of the plane with stable periodic points in a neighborhood of a homoclinic point

We consider self-diffeomorphisms of the plane of the class C ^r (1 ?? r < ??) with a fixed hyperbolic point and a nontransversal point homoclinic to it. We present a method for constructing a set of diffeomorphisms for which the neighborhood of a homoclinic point contains countably many stable periodic points with characteristic exponents bounded away from zero.     

Bifurcations of Morse-Smale diffeomorphisms with wildly embedded separatrices

We study bifurcations of Morse-Smale diffeomorphisms under a change of the embedding of the separatrices of saddle periodic points in the ambient 3-manifold. The results obtained are based on the following statement proved in this paper: for the 3-sphere, the space of diffeomorphisms of North Pole-South Pole type endowed with the C ¹ topology is connected. This statement is shown to be false in dimension 6.     

Quasi-morphismes de Calabi et graphe de Reeb sur le tore

We construct homogeneous quasi-morphisms on the identity component of the group of area preserving diffeomorphisms of the two dimensional torus, whose restriction to the subgroup of diffeomorphisms with support in any fixed disc equals Calabi's invariant. To cite this article: P. Py, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Resonance identity for symmetric closed characteristics on symmetric convex Hamiltonian energy hypersurfaces and its applications

In this paper, we establish a new resonance identity for symmetric closed characteristics on symmetric compact convex hypersurface Σ   in R²ⁿ${\mathbf{R}}^{2n}$ when there exist only finitely many geometrically distinct symmetric closed characteristics. As its applications, some interesting results about the stability and multiplicity of symmetric closed characteristics are obtained, and also we prove that if Σ   is C^∞$C^{\infty }$-generic, it carries infinitely many symmetric closed characteristics.     

Backward inducing and exponential decay of correlations for partially hyperbolic attractors

We study partially hyperbolic attractors ofC ² diffeomorphisms on a compact manifold. For a robust (non-empty interior) class of such diffeomorphisms, we construct Sinai-Ruelle-Bowen measures, for which we prove exponential decay of correlations and the central limit theorem, in the space of Hölder continuous functions. The techniques we develop (backward inducing, redundancy elimination algorithm) should be useful in the study of the stochastic properties of much more general non-uniformly hyperbolic systems.     

Dynamics on Ahlfors quasi-circles

The celebrated theory of Denjoy introduced a topological invariant distinguishingC ¹ andC ² diffeomorphisms of the circle. AC ² diffeomorphism of the circle cannot have an infinite minimal set other than the circle itself. However, this is possible forC ¹ diffeomorphisms. In dimension two there is a related invariant distinguishingC ² andC ³ diffeomorphisms. Partially supported by NSF grant No. MCS-83202062.     

Diffeomorphisms with <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-shadowing property

We give a characterization of structurally stable diffeomorphisms by making use of the notion of L ^p -shadowing property. More precisely, we prove that the set of structurally stable diffeomorphisms coincides with the C ¹-interior of the set of diffeomorphisms having L ^p -shadowing property.     

Diffeomorphisms with positive metric entropy

We obtain a dichotomy for \(C^{1}\)-generic, volume-preserving diffeomorphisms: either all the Lyapunov exponents of almost every point vanish or the volume is ergodic and non-uniformly Anosov (i.e. nonuniformly hyperbolic and the splitting into stable and unstable spaces is dominated). This completes a program first put forth by Ricardo Mañé.     

Une condition suffisante de minimalité pour les géodésiques de la métrique de HoferA sufficient minimality condition for geodesics of the Hofer's metric

We give a new sufficient condition for a Hamiltonian H to generate a length minimizing geodesic of the Hofer's metric on the group of Hamiltonian diffeomorphisms on $\text{R}{}^{\mathrm{2n}}$. This condition is related to the spectra of the linearized maps of the flow {φ^H_t} generated by H at the fixed points of the flow. In addition we show that if φ₀, φ₁ are two diffeomorphisms linked by such a geodesic, then the Hofer's distance between φ₀ and φ₁ is the same as Viterbo's one. To cite this article: J. Le Crapper, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 359–364.