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.     

Weakly Hyperbolic Actions of Kazhdan Groups on Tori

We study the ergodic and rigidity properties of weakly hyperbolic actions. First, we establish ergodicity for C² volume preserving weakly hyperbolic group actions on closed manifolds. For the integral action generated by a single Anosov diffeomorphism this theorem is classical and originally due to Anosov. Motivated by the Franks/Manning classification of Anosov diffeomorphisms on tori, we restrict our attention to weakly hyperbolic actions on the torus. When the acting group is a lattice subgroup of a semisimple Lie group with no compact factors and all (almost) simple factors of real rank at least two, we show that weak hyperbolicity in the original action implies weak hyperbolicity for the induced action on the fundamental group. As a corollary, we obtain that any such action on the torus is continuously semiconjugate to the affine action coming from the fundamental group via a map unique in the homotopy class of the identity. Under the additional assumption that some partially hyperbolic group element has quasi-isometrically embedded lifts of unstable leaves to the universal cover, we obtain a conjugacy, resulting in a continuous classification for these actions. Partially funded by VIGRE grant DMS-9977371 Received: January 2005 Revision: August 2005 Accepted: September 2005     

Embedding smooth diffeomorphisms in flows

In this paper we study the problem on embedding germs of smooth diffeomorphisms in flows in higher dimensional spaces. First we prove the existence of embedding vector fields for a local diffeomorphism with its nonlinear term a resonant polynomial. Then using this result and the normal form theory, we obtain a class of local Ck diffeomorphisms for k∈N∪{∞,ω} which admit embedding vector fields with some smoothness. Finally we prove that for any k∈N∪{∞} under the coefficient topology the subset of local Ck diffeomorphisms having an embedding vector field with some smoothness is dense in the set of all local Ck diffeomorphisms.     

Partial hyperbolicity far from homoclinic bifurcations

We prove that any diffeomorphism of a compact manifold can be C¹-approximated by a diffeomorphism which exhibits a homoclinic bifurcation (a homoclinic tangency or a heterodimensional cycle) or by a diffeomorphism which is partially hyperbolic (its chain-recurrent set splits into partially hyperbolic pieces whose centre bundles have dimensions less or equal to two). We also study in a more systematic way the central models introduced in Crovisier (in press) [10].     

Counting Closed Orbits of Hyperbolic Diffeomorphisms

In this paper we count closed orbits of a hyperbolic diffeomorphism restricted to a basic set. In fact we shall be dealing with two types of extensions of a hyperbolic diffeomorphism: a finite group extension (also called G-extensions) and also an automorphism extension (also called (G, ρ)-extensions). In particular we present Chebotarev type theorems and prime orbit theorem for such diffeomorphisms. These counting results are in the form of an asymptotic formula derived in complete analogy with the number theoretic result. The main difficulty is extending the domain of analyticity of the zeta and L-functions and this is overcome by resorting to symbolic dynamics. Unlike the case of a flow, the proof of this extension result does not rely on the properties of the Ruelle–Perron–Frobenius operator. Also the counting results do not use any tauberian theorems. Received: January 7, 2005.     

Center manifolds for nonuniformly partially hyperbolic diffeomorphisms

We establish the existence of smooth invariant center manifolds for the nonuniformly partially hyperbolic trajectories of a diffeomorphism in a Banach space. This means that the differentials of the diffeomorphism along the trajectory admit a nonuniform exponential trichotomy. We also consider the more general case of sequences of diffeomorphisms, which corresponds to a nonautonomous dynamics with discrete time. In addition, we obtain an optimal regularity for the center manifolds: if the diffeomorphisms are of class C^k then the manifolds are also of class C^k. As a byproduct of our approach we obtain an exponential control not only for the trajectories on the center manifolds, but also for their derivatives up to order k.     

Dynamical coherence of partially hyperbolic diffeomorphisms of tori isotopic to Anosov

We show that partially hyperbolic diffeomorphisms of \(d\) -dimensional tori isotopic to an Anosov diffeomorphism, where the isotopy is contained in the set of partially hyperbolic diffeomorphisms, are dynamically coherent. Moreover, we show a global stability result, i.e. every partially hyperbolic diffeomorphism as above is leaf-conjugate to the linear one. As a consequence, we obtain intrinsic ergodicity and measure equivalence for partially hyperbolic diffeomorphisms with one-dimensional center direction that are isotopic to Anosov diffeomorphisms through such a path.     

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.     

Conditions for a diffeomorghism to be embedded in aC r flow

We will consider aC ^r diffeomorphism of the real lineR, and give a necessary and sufficient condition for aC ^r diffeomorphism ofR to be embedded (uniquely) in aC ^r flow. As an application, we do the same for diffeomorphisms of the circleS ¹ and a class of analytic diffeomorphisms of the planeR ².     

SRB measures for partially hyperbolic systems whose central direction is mostly contracting

We consider partially hyperbolic diffeomorphisms preserving a splitting of the tangent bundle into a strong-unstable subbundleE ^uu (uniformly expanding) and a subbundleE ^c, dominated byE ^uu. We prove that if the central directionE ^c is mostly contracting for the diffeomorphism (negative Lyapunov exponents), then the ergodic Gibbsu-states are the Sinai-Ruelle-Bowen measures, there are finitely many of them, and their basins cover a full measure subset. If the strong-unstable leaves are dense, there is a unique Sinai-Ruelle-Bowen measure. We describe some applications of these results, and we also introduce a construction of robustly transitive diffeomorphisms in dimension larger than three, having no uniformly hyperbolic (neither contracting nor expanding) invariant subbundles. Work partially supported by CNRS and CNPq/PRONEX-Dynamical Systems, and carried out at Laboratoire de Topologie, Dijon, and IMPA, Rio de Janeiro.     

Statistical manifolds are statistical models

In this note we prove that any smooth (C¹ resp.) statistical manifold can be embedded into the space of probability measures on a finite set. As a result, we get positive answers to Lauritzen’s question and Amari’s question on a realization of smooth (C¹ resp.) statistical manifolds as finite dimensional statistical models.     

Separating diffeomorphisms of the torus

There exists a diffeomorphism on the n-dimensional torus Tⁿ which is conjugate with a hyperbolic linear automorphism, but is not an Anosov diffeomorphism. A diffeomorphismf: Tⁿ→Tⁿ has such a property iff is separating and belongs to the C⁰ closure of the Anosov diffeomorphisms.     

How often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency

Here we study an amazing phenomenon discovered by Newhouse [S. Newhouse, Non-density of Axiom A(a) on S², in: Proc. Sympos. Pure Math., vol. 14, Amer. Math. Soc., 1970, pp. 191-202; S. Newhouse, Diffeomorphisms with infinitely many sinks, Topology 13 (1974) 9-18; S. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets of diffeomorphisms, Publ. Math. Inst. Hautes Études Sci. 50 (1979) 101-151]. It turns out that in the space of Cr smooth diffeomorphisms Diffr(M) of a compact surface M there is an open set U such that a Baire generic diffeomorphism f∈U has infinitely many coexisting sinks. In this paper we make a step towards understanding “how often does a surface diffeomorphism have infinitely many sinks.” Our main result roughly says that with probability one for any positive D a surface diffeomorphism has only finitely many localized sinks either of cyclicity bounded by D or those whose period is relatively large compared to its cyclicity. It verifies a particular case of Palis' Conjecture saying that even though diffeomorphisms with infinitely many coexisting sinks are Baire generic, they have probability zero.One of the key points of the proof is an application of Newton Interpolation Polynomials to study the dynamics initiated in [V. Kaloshin, B. Hunt, A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms I, Ann. of Math., in press, 92 pp.; V. Kaloshin, A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms II, preprint, 85 pp.].     

Un critère d'existence d'invariant pour la conjugaison de difféomorphismes et de champs de vecteursAn existence criterion for conjugacy invariant of diffeomorphisms and vector fields

We consider dimension 3 vector fields (resp. diffeomorphisms), in a neighborhood of a hyperbolic saddle. We give a criterion to decide if the imaginary part (resp. the angular part) of the eigenvalues of the linear part of the dynamical system at the fixed point is a topological conjugacy invariant if we assume that the conjugacy maps a non-spiraling curve onto another one. We apply this result to the situation of diffeomorphisms and vector fields with a quasi-transversal homoclinic orbit. To cite this article: E. Dufraine, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 53–58     

Robust Weak Ergodicity And Stable Ergodicity

In this paper, we define robust weak ergodicity and study the relation between robust weak ergodicity and stable ergodicity for conservative partially hyperbolic systems. We prove that a C^r(r > 1) conservative partially hyperbolic diffeomorphism is stably ergodic if it is robustly weakly ergodic and has positive (or negative) central exponents on a positive measure set. Furthermore, if the condition of robust weak ergodicity is replaced by weak ergodicity, then the diffeomophism is an almost stably ergodic system. Additionally, we show in dimension three, a C^r(r > 1) conservative partially hyperbolic diffeomorphism can be approximated by stably ergodic systems if it is robustly weakly ergodic and robustly has non-zero central exponents.     

Nonexistence and partial existence of first integrals for diffeomorphisms

In this paper, we first present a necessary condition for diffeomorphisms to have formal first integrals. Moreover, we consider diffeomorphisms which have some nontrivial formal integrals. Under certain conditions, we show that the diffeomorphism does not admit any other formal integral in the sense of functionally independent first integrals.     

Stably ergodic diffeomorphisms which are not partially hyperbolic

We show stable ergodicity of a class of conservative diffeomorphisms ofT ⁿ which do not have any hyperbolic invariant subbundle. Moreover, the uniqueness of SRB (Sinai-Ruelle-Bowen) measure for non-conservativeC ¹ perturbations of such diffeomorphisms is verified. This class strictly contains non-partially hyperbolic robustly transitive diffeomorphisms constructed by Bonatti-Viana [4] and so we answer the question posed there on the stable ergodicity of such systems.     

Periodic shadowing and Ω-stability

We show that the following three properties of a diffeomorphism f of a smooth closed manifold are equivalent: (i) f belongs to the C ¹-interior of the set of diffeomorphisms having the periodic shadowing property; (ii) f has the Lipschitz periodic shadowing property; (iii) f is Ω-stable.     

Topological classification of gradient-like diffeomorphisms on 3-manifolds

We give a complete invariant, called global scheme, of topological conjugacy classes of gradient-like diffeomorphisms, on compact 3-manifolds. Conversely, we can realize any abstract global scheme by such a diffeomorphism.     

Smoothness of the carrying simplex for discrete-time competitive dynamical systems: A characterization of neat embedding

The paper is concerned with the question of smoothness of the carrying simplex S for a discrete-time dissipative competitive dynamical system. We give a necessary and sufficient criterion for S being a C¹ submanifold-with-corners neatly embedded in the nonnegative orthant, formulated in terms of inequalities between Lyapunov exponents for ergodic measures supported on the boundary of the orthant. This completes one thread of investigation occasioned by a question posed by M.W. Hirsch in 1988. Besides, amenable conditions are presented to guarantee the Cr (r?1) smoothness of S in the time-periodic competitive Kolmogorov systems of ODEs. Examples are also presented, one in which S is of class C¹ but not neatly embedded, the other in which S is not of class C¹.