Searcharxiv⌕ Search

arXiv subjects

Christian Bonatti

Publications and source records attributed to Christian Bonatti.

At least 55 records · Page 3Linked to original sources
arXiv2015

Robust criterion for the existence of nonhyperbolic ergodic measures

We give explicit $C^1$-open conditions that ensure that a diffeomorphism possesses a nonhyperbolic ergodic measure with positive entropy. Actually, our criterion provides the existence of a partially hyperbolic compact set with one-dimensional center and positive topological entropy on which the center Lyapunov exponent vanishes uniformly. The conditions of the criterion are met on a $C^1$-dense and open subset of the set of a diffeomorphisms having a robust cycle. As a corollary, there exists a $C^1$-open and dense subset of the set of non-Anosov robustly transitive diffeomorphisms consisting of systems with nonhyperbolic ergodic measures with positive entropy. The criterion is based on a notion of a blender defined dynamically in terms of strict invariance of a family of discs.

math.DS↗
arXiv2014

Partially hyperbolic diffeomorphisms with uniformly center foliation: the quotient dynamics

We show that a partially hyperbolic $C^1$ -diffeomorphism $f : M \to M$ with a uniformly compact $f$ -invariant center foliation $F^c$ is dynamically coherent. Further, the induced homeomorphism $F : M/F^c \to M/F^c$ on the quotient space of the center foliation has the shadowing property, i.e. for every $\varepsilon> 0$ there exists $δ> 0$ such that every $δ$-pseudo orbit of center leaves is $\varepsilon$-shadowed by an orbit of center leaves. Although the shadowing orbit is not necessarily unique, we prove the density of periodic center leaves inside the chain recurrent set of the quotient dynamics. Some other interesting properties of the quotient dynamics are discussed.

math.DS↗
arXiv2014

Building Anosov flows on 3-manifolds

We prove a result allowing to build (transitive or non-transitive) Anosov flows on 3-manifolds by gluing together filtrating neighborhoods of hyperbolic sets. We give several applications; for example: 1. we build a 3-manifold supporting both of a transitive Anosov vector field and a non-transitive Anosov vector field; 2. for any n, we build a 3-manifold M supporting at least n pairwise different Anosov vector fields; 3. we build transitive attractors with prescribed entrance foliation; in particular, we construct some incoherent transitive attractors; 4. we build a transitive Anosov vector field admitting infinitely many pairwise non-isotopic trans- verse tori.

math.DS↗
arXiv2013

Centralizers of $C^1$-contractions of the half line

A subgroup $G\subset Diff^1_+([0,1])$ is $C^1$-close to the identity if there is a sequence $h_n\in Diff^1_+([0,1])$ such that the conjugates $h_n g h_n^{-1}$ tend to the identity for the $C^1$-topology, for every $g\in G$. This is equivalent to the fact that $G$ can be embedded in the $C^1$-centralizer of a $C^1$-contraction of $[0,+\infty)$ (see [Fa] and Theorem 1.1). We first describe the topological dynamics of groups $C^1$-close to the identity. Then, we show that the class of groups $C^1$-close to the identity is invariant under some natural dynamical and algebraic extensions. As a consequence, we can describe a large class of groups $G\subset Diff^1_+([0,1])$ whose topological dynamics implies that they are $C^1$-close to the identity. This allows us to show that the free group ${\mathbb F}_2$ admits faithfull actions which are $C^1$-close to the identity. In particular, the $C^1$-centralizer of a $C^1$-contraction may contain free groups.

math.DS↗
arXiv2013

The $C^{1+α}$ hypothesis in Pesin theory revisited

We show that for every compact 3-manifold $M$ there exists an open subset of $\diff ^1(M)$ in which every generic diffeomorphism admits uncountably many ergodic probability measures which are hyperbolic while their supports are disjoint and admit a basis of attracting neighborhoods and a basis of repelling neighborhoods. As a consequence, the points in the support of these measures have no stable and no unstable manifolds. This contrasts with the higher regularity case, where Pesin theory gives us the stable and the unstable manifolds with complementary dimensions at almost every point. We also give such an example in dimension two, without local genericity.

math.DS↗
arXiv2013

Robust vanishing of all Lyapunov exponents for iterated function systems

Given any compact connected manifold $M$, we describe $C^2$-open sets of iterated functions systems (IFS's) admitting fully-supported ergodic measures whose Lyapunov exponents along $M$ are all zero. Moreover, these measures are approximated by measures supported on periodic orbits. We also describe $C^1$-open sets of IFS's admitting ergodic measures of positive entropy whose Lyapunov exponents along $M$ are all zero. The proofs involve the construction of non-hyperbolic measures for the induced IFS's on the flag manifold.

math.DS↗
arXiv2012

Flexible periodic points

We define the notion of $\varepsilon$-flexible periodic point: it is a periodic point with stable index equal to two whose dynamics restricted to the stable direction admits $\varepsilon$-perturbations both to a homothety and a saddle having an eigenvalue equal to one. We show that $\varepsilon$-perturbation to an $\varepsilon$-flexible point allows to change it in a stable index one periodic point whose (one dimensional) stable manifold is an arbitrarily chosen $C^1$ -curve. We also show that the existence of flexible point is a general phenomenon among systems with a robustly non-hyperbolic two dimensional center-stable bundle.

math.DS↗
arXiv2011

Tame dynamics and robust transitivity

One main task of smooth dynamical systems consists in finding a good decomposition into elementary pieces of the dynamics. This paper contributes to the study of chain-recurrence classes. It is known that $C^1$-generically, each chain-recurrence class containing a periodic orbit is equal to the homoclinic class of this orbit. Our result implies that in general this property is fragile. We build a C1-open set U of tame diffeomorphisms (their dynamics only splits into finitely many chain-recurrence classes) such that for any diffeomorphism in a C-infinity-dense subset of U, one of the chain-recurrence classes is not transitive (and has an isolated point). Moreover, these dynamics are obtained among partially hyperbolic systems with one-dimensional center. Résumé : Dynamique modérée et transitivité robuste. L'un des buts des systèmes dynamiques consiste à trouver une bonne décomposition de la dynamique en pièces élémentaires. Cet article contribue à l'étude des classes de récurrence par chaînes. On sait que C1-génériquement, chaque classe de récurrence par chaînes contenant une orbite périodique coincide avec la classe homocline de cette orbite. Notre résultat montre que cette propriété est en générale fragile. Nous construisons un ouvert U de difféomorphismes modérés (leur dynamique ne se décompose qu'en un nombre fini de classes de récurrence par chaînes) tel que pour tout difféomorphisme appartenant à un sous-ensemble C-infini-dense de U, une des classes de récurrence par chaînes n'est pas transitive (elle a un point isolé). De plus, ces dynamiques sont obtenues comme systèmes partiellement hyperboliques avec une direction centrale uni-dimensionnelle.

math.DS↗
arXiv2011

Perturbation of the Lyapunov spectra of periodic orbits

We describe all Lyapunov spectra that can be obtained by perturbing the derivatives along periodic orbits of a diffeomorphism. The description is expressed in terms of the finest dominated splitting and Lyapunov exponents that appear in the limit of a sequence of periodic orbits, and involves the majorization partial order. Among the applications, we give a simple criterion for the occurrence of universal dynamics.

math.DS↗
arXiv2011

Stabilization of heterodimensional cycles

We consider diffeomorphisms $f$ with heteroclinic cycles associated to saddles $P$ and $Q$ of different indices. We say that a cycle of this type can be stabilized if there are diffeomorphisms close to $f$ with a robust cycle associated to hyperbolic sets containing the continuations of $P$ and $Q$. We focus on the case where the indices of these two saddles differ by one. We prove that, excluding one particular case (so-called twisted cycles that additionally satisfy some geometrical restrictions), all such cycles can be stabilized.

math.DS↗
arXiv2011

Fragile cycles

We study diffeomorphisms $f$ with heterodimensional cycles, that is, heteroclinic cycles associated to saddles $p$ and $q$ with different indices. Such a cycle is called fragile if there is no diffeomorphism close to $f$ with a robust cycle associated to hyperbolic sets containing the continuations of $p$ and $q$. We construct a codimension one submanifold of $\Difr(\SS^2\times \SS^1)$, $r\ge 1$, that consists of diffeomorphisms with fragile heterodimensional cycles. Our construction holds for any manifold of dimension $\ge 4$.

math.DS↗
arXiv2009

On the existence of attractors

On every compact 3-manifold, we build a non-empty open set $\cU$ of $\Diff^1(M)$ such that, for every $r\geq 1$, every $C^r$-generic diffeomorphism $f\in\cU\cap \Diff^r(M)$ has no topological attractors. On higher dimensional manifolds, one may require that $f$ has neither topological attractors nor topological repellers. Our examples have finitely many quasi attractors. For flows, we may require that these quasi attractors contain singular points. Finally we discuss alternative definitions of attractors which may be better adapted to generic dynamics.

math.DS↗
arXiv2008

Nonuniform hyperbolicity for C^1-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 nonuniformly hyperbolic: they exhibit no zero Lyapunov exponents. Third, we extend a theorem by Sigmund on hyperbolic basic sets: every isolated transitive set L of any C^1-generic diffeomorphism f exhibits many ergodic hyperbolic measures whose supports coincide with the whole set L. In addition, confirming a claim made by R. Mané in 1982, we show that hyperbolic measures whose Oseledets splittings are dominated satisfy Pesin's Stable Manifold Theorem, even if the diffeomorphism is only C^1.

math.DS↗