Structural stability of C1 diffeomorphisms

In this paper we prove that if f is a C¹ diffeomorphism that satisfies Axiom A and the strong transversality condition then it is structurally stable. J. Robbin proved this theorem for C² diffeomorphisms. In addition to reducing the amount of differentiability necessary to prove the theorem, we also give a new proof combining the d_f metric of Robbin with the stable and unstable manifold proof of D. Anosov. We also prove structural stability in the neighborhood of a single hyperbolic basic set (independent of its being part of a diffeomorphism that satisfies Axiom A and the strong transversality condition). These proofs are adapted to prove the structural stability of C¹ flows in another paper.     

A λ-lemma for foliations

We show that a C⁰ codimension one foliation with C¹ leaves F of a closed manifold is minimal if there are a foliation G transverse to F, and a diffeomorphism f preserving both foliations, such that every leaf of F intersects every leaf of G and f expands G. We use this result to study of Anosov actions on closed manifolds.     

On manifolds supporting quasi-Anosov diffeomorphisms

Let M be an n-dimensional manifold supporting a quasi-Anosov diffeomorphism. If n=3 then either $\text{M=}\text{T}{}^3$, in which case the diffeomorphisms is Anosov, or else its fundamental group contains a copy of $\text{Z}{}^6$. If n=4 then Π₁(M) contains a copy of $\text{Z}{}^4$, provided that the diffeomorphism is not Anosov. To cite this article: J. Rodriguez Hertz et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 321–323.     

Three dimensional expansive diffeomorphisms with homoclinic points

LetM be a compact connected oriented three dimensional manifold andf:MM an expansive diffeomorphism such that (f)=M. Let us also assume that there is a hyperbolic periodic point with a homoclinic intersection. Thenf is conjugate to an Anosov isomorphism ofT ³. Moreover, we show that at a homoclinic point the stable and unstable manifolds of the hyperbolic periodic point are topologically transverse.     

A <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> Anosov diffeomorphism with a horseshoe that attracts almost any point

We present an example of a C ¹ Anosov diffeomorphism with a physical measure such that its basin has full Lebesgue measure and its support is a horseshoe of zero measure.     

On Surface Attractors and Repellers in 3-Manifolds

We show that if f: M ³ → M ³ is an A diffeomorphism with a surface two-dimensional attractor or repeller $\mathcal{B}$ with support $M_\mathcal{B}^2$ , then $\mathcal{B} = M_\mathcal{B}^2$ and there exists a k ≥ 1 such that (1) $M_\mathcal{B}^2$ is the disjoint union M ₁ ² ? ? ? M _k ² of tame surfaces such that each surface M _i ² is homeomorphic to the 2-torus T ²; (2) the restriction of f ^k to M _i ² , i ∈ {1,..., k}, is conjugate to an Anosov diffeomorphism of the torus T ².     

Dehn twists and pseudo-Anosov diffeomorphisms

Summary We show that it is possible to obtain many pseudo-Anosov diffeomorphisms from Dehn twists. In particular, we generalize a theorem of Long and Morton to obtain that iff is a pseudo-Anosov diffeomorphism of an oriented surface andT is the Dehn twist around the simple closed curve , then the isotopy class ofT ⁿ f contains a pseudo-Anosov diffeomorphism except for at most 7 consecutive values ofn.     

Transitive partially hyperbolic diffeomorphisms on 3-manifolds

The known examples of transitive partially hyperbolic diffeomorphisms on 3-manifolds belong to 3 basic classes: perturbations of skew products over an Anosov map of T², perturbations of the time one map of a transitive Anosov flow, and certain derived from Anosov diffeomorphisms of the torus T³. In this work we characterize the two first types by a local hypothesis associated to one closed periodic curve.     

On C1-class local diffeomorphisms whose periodic points are nonuniformly expanding

Using a sifting-shadowing combination, we prove in this paper that an arbitrary C¹-class local diffeomorphism f of a closed manifold M ⁿ is uniformly expanding on the closure ${\mathrm{Cl}_{M^n}(\mathrm{Per}(f))}$ of its periodic point set Per(f), if it is nonuniformly expanding on Per(f).     

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.     

On the approximation of time one maps of Anosov flows by Axiom A diffeomorphisms

We prove that if 𝒻₁ is the time one map of a transitive and codimension one Anosov flow φ and it is C ¹-approximated by Axiom A diffeomorphisms satisfying a property called P, then the flow is topologically conjugated to the suspension of a codimension one Anosov diffeomorphism. A diffeomorphism 𝒻 satisfies property P if for every periodic point in M the number of periodic points in a fundamental domain of its central manifold is constant. Received: 15 March 2001     

On the dynamical coherence of structurally stable 3-diffeomorphisms

We prove that each structurally stable diffeomorphism f on a closed 3-manifold M ³ with a two-dimensional surface nonwandering set is topologically conjugated to some model dynamically coherent diffeomorphism.     

The barycenter property for robust and generic diffeomorphisms

Let f:M~d→M~d(d≥2) be a diffeomorphism on a compact C~∞ manifold on M.If a diffeomorphism f belongs to the C~1-interior of the set of all diffeomorphisms having the barycenter property,then f is Ω-stable.Moreover,if a generic diffeomorphism f has the barycenter property,then f is Ω-stable.We also apply our results to volume preserving diffeomorphisms.     

On the density of Birkhoff sums for Anosov diffeomorphisms

Let f : M → M be an Anosov diffeomorphism on a nilmanifold. We consider Birkhoff sums for a Holder continuous observation along periodic orbits. We show that if there are two Birkhoff sums distributed at both sides of zero, then the set of Birkhoff sums of all the periodic points is dense in R.     

Limit laws of entrance times for homeomorphisms of the circle

Given a homeomorphismf of the circle with irrational rotation number and a descending chain of renormalization intervalsj _n off, we consider for each interval the point process obtained by marking the times for the orbit of a point in the circle to enterJ _n. Assuming the point is randomly chosen by the unique invariant probability measure off, we obtain necessary and sufficient conditions which guarantee convergence in law of the corresponding point process and we describe all the limiting processes. These conditions are given in terms of the convergent subsequences of the orbit of the rotation number off under the Gauss transformation and under a certain realization of its natural extension. We also consider the case when the point is randomly chosen according to Lebesgue measure,f being a diffeomorphism which isC ¹-conjugate to a rotation, and we show that the same necessary and sufficient conditions guarantee convergence in this case.     

A boundary uniqueness theorem for holomorphic functions of several complex variables

If D ? Cⁿ is a region with a smooth boundary and M ? ?D is a smooth manifold such that for some point p ∈ M the complex linear hull of the tangent plane T_p(M) coincides with Cⁿ, then for each functionf ε A(D) the conditionf¦M=0 implies thatf=0 in D.     

Constructing infra-nilmanifolds admitting an Anosov diffeomorphism

In this paper we establish an algebraic characterization of those infra-nilmanifolds modeled on a free c-step nilpotent Lie group and with an abelian holonomy group admitting an Anosov diffeomorphism. We also develop a new method for constructing examples of infra-nilmanifolds having an Anosov diffeomorphism.     

An example of convex Hamiltonian diffeomorphism where asymptotic distance from identity is strictly greater than the minimal action

We construct a convex Hamiltonian diffeomorphism on the unit ball of cotangent bundle of Tn (n?2), where the asymptotic distance from identity is strictly greater than the minimal action.     

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.