Conservative Diffeomorphisms Research Articles

Topological pressure for conservative C 1-diffeomorphisms with no dominated splitting

We prove three formulas for computing the topological pressure of C 1 -generic conservative diffeomorphism with no dominated splitting and show the continuity of topological pressure with respect to these diffeomorphisms. We prove for these generic diffeomorphisms that there are no equilibrium states with positive measure theoretic entropy. In particular, for hyperbolic potentials, there are no equilibrium states. For C 1 generic conservative diffeomorphism on compact surfaces with no dominated splitting and ϕ m ( x ) := − 1 m log ⁡ ‖ D x f m ‖ , m ∈ N , we show that there exist equilibrium states with zero entropy and there exists a transition point t 0 for the one parameter family { t ϕ m } t ≥ 0 , such that there is no equilibrium states for t ∈ [ 0 , t 0 ) and there is an equilibrium state for t ∈ [ t 0 , + ∞ ) .

Homoclinic orbits for area preserving diffeomorphisms of surfaces

AbstractWe show that $C^r $ generically in the space of $C^r$ conservative diffeomorphisms of a compact surface, every hyperbolic periodic point has a transverse homoclinic orbit.

On mixing diffeomorphisms of the disc

We prove that a $$C^k$$, $$k\ge 2$$ pseudo-rotation f of the disc with non-Brjuno rotation number is $$C^{k-1}$$-rigid. The proof is based on two ingredients: (1) we derive from Franks’ Lemma on free discs that a pseudo-rotation with small rotation number compared to its $$C^1$$ norm must be close to the identity map; (2) using Pesin theory, we obtain an effective finite information version of the Katok closing lemma for an area preserving surface diffeomorphism f, that provides a controlled gap in the possible growth of the derivatives of f between exponential and sub-exponential. Our result on rigidity, together with a KAM theorem by Russmann, allow to conclude that analytic pseudo-rotations of the disc or the sphere are never topologically mixing. Due to a structure theorem by Franks and Handel of zero entropy surface diffeomorphisms, it follows that an analytic conservative diffeomorphism of the disc or the sphere that is topologically mixing must have positive topological entropy.

On Herman's positive entropy conjecture

We show that any area-preserving Cr-diffeomorphism of a two-dimensional surface displaying an elliptic fixed point can be Cr-perturbed to one exhibiting a chaotic island whose metric entropy is positive, for every 1≤r≤∞. This proves a conjecture of Herman stating that the identity map of the disk can be C∞-perturbed to a conservative diffeomorphism with positive metric entropy. This implies also that the Chirikov standard map for large and small parameter values can be C∞-approximated by a conservative diffeomorphisms displaying a positive metric entropy (a weak version of Sinai's positive metric entropy conjecture). Finally, this sheds light onto a Herman's question on the density of Cr-conservative diffeomorphisms displaying a positive metric entropy: we show the existence of a dense set formed by conservative diffeomorphisms which either are weakly stable (so, conjecturally, uniformly hyperbolic) or display a chaotic island of positive metric entropy.

Hyperbolicity for Conservative Diffeomorphisms

This paper introduces the notion of robust hyperbolicity along periodic orbits homoclinically related to p (RNUHP) for conservative diffeomorphisms. It is proved that if f ∈ Diff1+ m (M) is RNUHP, then f is Anosov. It is also shown that f admits a dominated splitting, provided that f is expansive conservative stable.

The 𝐶¹ density of nonuniform hyperbolicity in 𝐶^{𝑟} conservative diffeomorphisms

Let D i f f m r ( M ) \mathrm {Diff}^{r}_m(M) be the set of C r C^{r} volume-preserving diffeomorphisms on a compact Riemannian manifold M M ( dim ⁡ M ≥ 2 \dim M\geq 2 ). In this paper, we prove that the diffeomorphisms without zero Lyapunov exponents on a set of positive volume are C 1 C^1 dense in D i f f m r ( M ) , r ≥ 1 \mathrm {Diff}^{r}_m(M), r\geq 1 . We also prove a weaker result for symplectic diffeomorphisms S y m ω r ( M ) , r ≥ 1 \mathrm {Sym}^{r}_{\omega }(M), r\geq 1 saying that either the symplectic diffeomorphism f f is partially hyperbolic or C 1 C^1 arbitrarily close to f f in S y m ω r ( M ) \mathrm {Sym}^{r}_{\omega }(M) , there is a diffeomorphism g ∈ S y m ω r ( M ) g\in \mathrm {Sym}^{r}_{\omega }(M) without zero Lyapunov exponents on a set with positive volume.

Transitivity of conservative diffeomorphisms isotopic to Anosov on

We prove transitivity for volume-preserving $C^{1+}$ diffeomorphisms on $\mathbb{T}^{3}$ which are isotopic to a linear Anosov automorphism along a path of weakly partially hyperbolic diffeomorphisms.

Some results on perturbations of Lyapunov exponents

In this paper, we study two properties of the Lyapunov exponents under small perturbations: one is when we can remove zero Lyapunov exponents and the other is when we can distinguish all the Lyapunov exponents. The first result shows that we can perturb all the zero integrated Lyapunov exponents $\int_M \lambda_j(x)d\omega(x)$ into nonzero ones, for any partially hyperbolic diffeomorphism. The second part contains an example which shows the local genericity of diffeomorphisms with non-simple spectrum and three results: one discusses the relation between simple-spectrum property and the existence of complex eigenvalues; the other two describe the difference on the spectrum between the diffeomorphisms far from homoclinic tangencies and those in the interior of the complement. Moreover, among the conservative diffeomorphisms far from tangencies, we prove that ergodic ones form a residual subset.

Genericity of nonuniform hyperbolicity in dimension 3

For a generic conservative diffeomorphism of a closed connected 3-manifold $M$, the Oseledets splitting is a globally dominated splitting. Moreover, either all Lyapunov exponents vanish almost everywhere, or else the system is nonuniformly hyperbolic and ergodic. &nbsp This is the 3-dimensional version of the well-known result by Mane-Bochi [14, 4], stating that a generic conservative surface diffeomorphism is either Anosov or all Lyapunov exponents vanish almost everywhere. This result was inspired by and answers in the positive in dimension 3 a conjecture by Avila-Bochi [2].

Creation of blenders in the conservative setting

In this work we prove that each Cr conservative diffeomorphism with a pair of hyperbolic periodic points of co-index one can be C1-approximated by Cr conservative diffeomorphisms having a blender.

$C^1$-generic conservative diffeomorphisms have trivial centralizer

We prove that the spaces of $C^1$ symplectomorphisms and of $C^1$ volume-preserving diffeomorphisms of connected manifolds contain residual subsets of diffeomorphisms whose centralizers are trivial.

Equivalent Conditions of Dominated Splitting for Volume-Preserving Diffeomorphisms

We discuss the equivalent conditions of dominated splitting for conservative diffeomorphisms in C1 topology.

A remark on conservative diffeomorphisms

We show that a stably ergodic diffeomorphism can be C 1 approximated by a diffeomorphism having stably non-zero Lyapunov exponents. To cite this article: J. Bochi et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).

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.

Robust transitivity and almost robust ergodicity

In this paper, we prove a relation between robust transitivity and robust ergodicity for conservative diffeomorphisms. In dimension two, robustly transitive diffeomorphisms are robustly ergodic. In higher dimensions, we work with partially hyperbolic diffeomorphisms with non-zero Lyapunov exponents. We define almost robust ergodicity and prove that in the three-dimensional case robustly transitive diffeomorphisms are approximated by almost robustly ergodic ones. In higher dimensions under the condition of the central Lyapunov exponents being of the same sign, we prove that robustly transitive diffeomorphisms are robustly ergodic.

Recurrence and genericity

We prove a C 1-connecting lemma for pseudo-orbits of diffeomorphisms on compact manifolds. We explore some consequences for C 1-generic diffeomorphisms. For instance, C 1-generic conservative diffeomorphisms are transitive. To cite this article: C. Bonatti, S. Crovisier, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Plenty of elliptic islands for the standard family of area preserving maps

For the Standard Map, a well-known family of conservative diffeomorphisms on the torus, we construct large basic sets which fill in the torus as the parameter runs to ∞. Then we prove that, for a residual set of large parameters, these basic sets are accumulated by elliptic periodic islands. We also show that there exists a k0 > 0 and a dense set of parameters in [k0, ∞) for which the standard map exhibits homoclinic tangencies.

Transversal homoclinic points of a class of conservative diffeomorphisms

We develop a global graph transformation to obtain estimates for certain invariant manifolds of a class of area preserving diffeomorphisms with symmetries. The existence of homoclinic or heteroclinic points and the analyticity of the angle between the invariant manifolds at them is proved for the Hénon and the standard maps. A lower bound of this angle is given for a large set of values of the parameters.

Examples of conservative diffeomorphisms of the two-dimensional torus with coexistence of elliptic and stochastic behaviour

AbstractWe find very simple examples of C∞-arcs of diffeomorphisms of the two-dimensional torus, preserving the Lebesgue measure and having the following properties: (1) the beginning of an arc is inside the set of Anosov diffeomorphisms; (2) after the bifurcation parameter every diffeomorphism has an elliptic fixed point with the first Birkhoff invariant non-zero (the KAM situation) and an invariant open area with almost everywhere non-zero Lyapunov characteristic exponents, moreover where the diffeomorphism has Bernoulli property; (3) the arc is real-analytic except on two circles (for each value of parameter) which are inside the Bernoulli property area.