Attracteurs de Lorenz de variété instable de dimension arbitraire

Abstract We construct the first examples of flows with robust multidimensional Lorenz-like attractors: the singularity contained in the attractor may have any number of expanding eigenvalues, and the attractor remains transitive in a whole neighbourhood of the initial flow. These attractors support a Sinai-Ruelle-Bowen SRB-measure and, contrary to the usual (low-dimensional) Lorenz models, they have infinite modulus of structural stability.

Transitive Anosov flows and Axiom-A diffeomorphisms

AbstractLet M be a smooth compact Riemannian manifold without boundary, and ϕ:M×ℝ→M a transitive Anosov flow. We prove that if the time-one map of ϕ is C1-approximated by Axiom-A diffeomorphisms with more than one attractor, then ϕ is topologically equivalent to the suspension of an Anosov diffeomorphism.

Small $C^1$ actions of semidirect products on compact manifolds

Let $T$ be a compact fibered $3$--manifold, presented as a mapping torus of a compact, orientable surface $S$ with monodromy $\psi$, and let $M$ be a compact Riemannian manifold. Our main result is that if the induced action $\psi^*$ on $H^1(S,\mathbb{R})$ has no eigenvalues on the unit circle, then there exists a neighborhood $\mathcal U$ of the trivial action in the space of $C^1$ actions of $\pi_1(T)$ on $M$ such that any action in $\mathcal{U}$ is abelian. We will prove that the same result holds in the generality of an infinite cyclic extension of an arbitrary finitely generated group $H$, provided that the conjugation action of the cyclic group on $H^1(H,\mathbb{R})\neq 0$ has no eige…

The centralizer of a C1 generic diffeomorphism is trivial

In this announcement, we describe the solution in the C1 topology to a question asked by S. Smale on the genericity of trivial centralizers: the set of diffeomorphisms of a compact connected manifold with trivial centralizer residual in Diff^1 but does not contain an open and dense subset.

Roots in the mapping class groups

The purpose of this paper is the study of the roots in the mapping class groups. Let $\Sigma$ be a compact oriented surface, possibly with boundary, let $\PP$ be a finite set of punctures in the interior of $\Sigma$, and let $\MM (\Sigma, \PP)$ denote the mapping class group of $(\Sigma, \PP)$. We prove that, if $\Sigma$ is of genus 0, then each $f \in \MM (\Sigma)$ has at most one $m$-root for all $m \ge 1$. We prove that, if $\Sigma$ is of genus 1 and has non-empty boundary, then each $f \in \MM (\Sigma)$ has at most one $m$-root up to conjugation for all $m \ge 1$. We prove that, however, if $\Sigma$ is of genus $\ge 2$, then there exist $f,g \in \MM (\Sigma, \PP)$ such that $f^2=g^2$, $…

Existence of common zeros for commuting vector fields on 3‐manifolds II. Solving global difficulties

We address the following conjecture about the existence of common zeros for commuting vector fields in dimension three: if $X,Y$ are two $C^1$ commuting vector fields on a $3$-manifold $M$, and $U$ is a relatively compact open such that $X$ does not vanish on the boundary of $U$ and has a non vanishing Poincar\'e-Hopf index in $U$, then $X$ and $Y$ have a common zero inside $U$. We prove this conjecture when $X$ and $Y$ are of class $C^3$ and every periodic orbit of $Y$ along which $X$ and $Y$ are collinear is partially hyperbolic. We also prove the conjecture, still in the $C^3$ setting, assuming that the flow $Y$ leaves invariant a transverse plane field. These results shed new light on t…

Periodic measures and partially hyperbolic homoclinic classes

In this paper, we give a precise meaning to the following fact, and we prove it: $C^1$-open and densely, all the non-hyperbolic ergodic measures generated by a robust cycle are approximated by periodic measures. We apply our technique to the global setting of partially hyperbolic diffeomorphisms with one dimensional center. When both strong stable and unstable foliations are minimal, we get that the closure of the set of ergodic measures is the union of two convex sets corresponding to the two possible $s$-indices; these two convex sets intersect along the closure of the set of non-hyperbolic ergodic measures. That is the case for robustly transitive perturbation of the time one map of a tr…

Tame dynamics and robust transitivity chain-recurrence classes versus homoclinic classes

Anomalous Anosov flows revisited

This paper is devoted to higher dimensional Anosov flows and consists of two parts. In the first part, we investigate fiberwise Anosov flows on affine torus bundles which fiber over 3-dimensional Anosov flows. We provide a dichotomy result for such flows --- they are either suspensions of Anosov diffeomorphisms or the stable and unstable distributions have equal dimensions. In the second part, we give a new surgery type construction of Anosov flows, which yields non-transitive Anosov flows in all odd dimensions.

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.

Abnormal escape rates from nonuniformly hyperbolic sets

Consider a $C^{1+\epsilon}$ diffeomorphism $f$ having a uniformly hyperbolic compact invariant set $\Omega$, maximal invariant in some small neighbourhood of itself. The asymptotic exponential rate of escape from any small enough neighbourhood of $\Omega$ is given by the topological pressure of $-\log |J^u f|$ on $\Omega$ (Bowen–Ruelle in 1975). It has been conjectured (Eckmann–Ruelle in 1985) that this property, formulated in terms of escape from the support $\Omega$ of a (generalized Sinai–Ruelle–Bowen (SRB)) measure, using its entropy and positive Lyapunov exponents, holds more generally. We present a simple $C^\infty$ two-dimensional counterexample, constructed by a surgery using a Bowe…

Feuilletages deCP(n) : de l’holonomie hyperbolique pour les minimaux exceptionnels

Let ℱ be a holomorphic foliation ofCP(n). If ℱ has a leaf L, the closure L of which is disjoint from the singular set of the foliation, we prove that there exists a loop in a leaf contained in L with contracting hyperbolic holonomy.

A mechanism for ejecting a horseshoe from a partially hyperbolic chain recurrence class

We give a $C^1$-perturbation technique for ejecting an a priori given finite set of periodic points preserving a given finite set of homo/hetero-clinic intersections from a chain recurrence class of a periodic point. The technique is first stated under a simpler setting called Markov iterated function system, a two dimensional iterated function system in which the compositions are chosen in Markovian way. Then we apply the result to the setting of three dimensional partially hyperbolic diffeomorphisms.

Robust existence of nonhyperbolic ergodic measures with positive entropy and full support

We prove that for some manifolds $M$ the set of robustly transitive partially hyperbolic diffeomorphisms of $M$ with one-dimensional nonhyperbolic centre direction contains a $C^1$-open and dense subset of diffeomorphisms with nonhyperbolic measures which are ergodic, fully supported and have positive entropy. To do so, we formulate abstract conditions sufficient for the construction of an ergodic, fully supported measure $\mu$ which has positive entropy and is such that for a continuous function $\phi\colon X\to\mathbb{R}$ the integral $\int\phi\,d\mu$ vanishes. The criterion is an extended version of the control at any scale with a long and sparse tail technique coming from the previous w…

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.

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.

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.

Hasse diagrams and orbit class spaces

Abstract Let X be a topological space and G be a group of homeomorphisms of X. Let G ˜ be an equivalence relation on X defined by x G ˜ y if the closure of the G-orbit of x is equal to the closure of the G-orbit of y. The quotient space X / G ˜ is called the orbit class space and is endowed with the natural order inherited from the inclusion order of the closure of the classes, so that, if such a space is finite, one can associate with it a Hasse diagram. We show that the converse is also true: any finite Hasse diagram can be realized as the Hasse diagram of an orbit class space built from a dynamical system ( X , G ) where X is a compact space and G is a finitely generated group of homeomo…

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. Nous montrons un lemme de connexion C^1 pour les pseudo-orbites des diffeomorphismes des varietes compactes. Nous explorons alors les consequences pour les diffeomorphismes C^1-generiques. Par exemple, les diffeomorphismes conservatifs C^1-generiques sont transitifs.

Anomalous partially hyperbolic diffeomorphisms I: dynamically coherent examples

We build an example of a non-transitive, dynamically coherent partially hyperbolic diffeomorphism $f$ on a closed $3$-manifold with exponential growth in its fundamental group such that $f^n$ is not isotopic to the identity for all $n\neq 0$. This example contradicts a conjecture in \cite{HHU}. The main idea is to consider a well-understood time-$t$ map of a non-transitive Anosov flow and then carefully compose with a Dehn twist.

Generic Properties of Dynamical Systems

The state of a concrete system (from physics, chemistry, ecology, or other sciences) is described using (finitely many, say n) observable quantities (e.g., positions and velocities for mechanical systems, population densities for echological systems, etc.). Hence, the state of a system may be represented as a point $x$ in a geometrical space $\mathbb R^n$. In many cases, the quantities describing the state are related, so that the phase space (space of all possible states) is a submanifold $M\subset \mathbb R^n$. The time evolution of the system is represented by a curve $x_t$, $t \in\mathbb R$ drawn on the phase space $M$, or by a sequence $x_n \in M$, $n \in\mathbb Z$, if we consider disc…

Surface homeomorphisms with zero dimensional singular set

We prove that if f is an orientation-preserving homeomorphism of a closed orientable surface M whose singular set is totally disconnected, then f is topologically conjugate to a conformal transformation.

SURVEY Towards a global view of dynamical systems, for the C1-topology

AbstractThis paper suggests a program for getting a global view of the dynamics of diffeomorphisms, from the point of view of the C1-topology. More precisely, given any compact manifold M, one splits Diff1(M) into disjoint C1-open regions whose union is C1-dense, and conjectures state that each of these open sets and their complements is characterized by the presence of: •either a robust local phenomenon;•or a global structure forbidding this local phenomenon. Other conjectures state that some of these regions are empty. This set of conjectures draws a global view of the dynamics, putting in evidence the coherence of the numerous recent results on C1-generic dynamics.

Partially hyperbolic diffeomorphisms with a uniformly compact center foliation: the quotient dynamics

We show that a partially hyperbolic$C^{1}$-diffeomorphism$f:M\rightarrow M$with a uniformly compact$f$-invariant center foliation${\mathcal{F}}^{c}$is dynamically coherent. Further, the induced homeomorphism$F:M/{\mathcal{F}}^{c}\rightarrow M/{\mathcal{F}}^{c}$on the quotient space of the center foliation has the shadowing property, i.e. for every${\it\epsilon}>0$there exists${\it\delta}>0$such that every${\it\delta}$-pseudo-orbit of center leaves is${\it\epsilon}$-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. Other interesting proper…

Un exemple de flot d'Anosov transitif transverse à un tore et non conjugué à une suspension

AbstractWe construct an example of transitive Anosov flow on a compact 3-manifold, which admits a transversal torus and is not the suspension of an Anosov diffeomorphism.

Perturbations of the derivative along periodic orbits

International audience; We show that a periodic orbit of large period of a diffeomorphism or flow, either admits a dominated splitting of a prescribed strength, or can be turned into a sink or a source by a C1-small perturbation along the orbit. As a consequence we show that the linear Poincaré flow of a C1-vector field admits a dominated splitting over any robustly transitive set.

A criterion for zero averages and full support of ergodic measures

International audience; Consider a homeomorphism $f$ defined on a compact metric space $X$ and a continuous map $\phi\colon X \to \mathbb{R}$. We provide an abstract criterion, called control at any scale with a long sparse tail for a point $x\in X$ and the map $\phi$, which guarantees that any weak* limit measure $\mu$ of the Birkhoff average of Dirac measures $\frac1n\sum_0^{n-1}\delta(f^i(x))$ s such that $\mu$-almost every point $y$ has a dense orbit in $X$ and the Birkhoff average of $\phi$ along the orbit of $y$ is zero.As an illustration of the strength of this criterion, we prove that the diffeomorphisms with nonhyperbolic ergodic measures form a $C^1$-open and dense subset of the s…

Non-wandering sets with non-empty interiors

We study diffeomorphisms of a closed connected manifold whose non-wandering set has a non-empty interior and conjecture that C1-generic diffeomorphisms whose non-wandering set has a non-empty interior are transitive. We prove this conjecture in three cases: hyperbolic diffeomorphisms, partially hyperbolic diffeomorphisms with two hyperbolic bundles, and tame diffeomorphisms (in the first case, the conjecture is folklore; in the second one, it follows by adapting the proof in Brin (1975 Topological transitivity of a certain class of dynamical systems, and flows of frames on manifolds of negative curvature Funct. Anal. Appl. 9 9–19)).We study this conjecture without global assumptions and pro…

3-manifolds which are orbit spaces of diffeomorphisms

Abstract In a very general setting, we show that a 3-manifold obtained as the orbit space of the basin of a topological attractor is either S 2 × S 1 or irreducible. We then study in more detail the topology of a class of 3-manifolds which are also orbit spaces and arise as invariants of gradient-like diffeomorphisms (in dimension 3). Up to a finite number of exceptions, which we explicitly describe, all these manifolds are Haken and, by changing the diffeomorphism by a finite power, all the Seifert components of the Jaco–Shalen–Johannson decomposition of these manifolds are made into product circle bundles.

The index of stable critical points

Abstract In this paper we show that in dimension greater or equal than 3 the index of a stable critical point can be any integer. More concretely, given any k∈ Z and n⩾3 we construct a C ∞ vector field on R n with a unique critical point which is stable (in positive and negative time) and has index equal to k. This result extends previous ones on the index of stable critical points.

A C1-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources

We show that, for every compact n-dimensional manifold, n > 1, there is a residual subset of Diff (M) of diffeomorphisms for which the homoclinic class of any periodic saddle of f verifies one of the following two possibilities: Either it is contained in the closure of an infinite set of sinks or sources (Newhouse phenomenon), or it presents some weak form of hyperbolicity called dominated splitting (this is a generalization of a bidimensional result of Mafine [Ma3]). In particular, we show that any Cl-robustly transitive diffeomorphism admits a dominated splitting.

Flots de Smale en dimension 3: présentations finies de voisinages invariants d'ensembles selles

Abstract Given a vector field X on a compact 3-manifold, and a hyperbolic saddle-like set K of that vector field, we consider all the filtering neighbourhood of K: by such, we mean any submanifold which boundary is tranverse to X, the maximal invariant of which is equal to K and which intersection with every orbit of X is connected. Up to topological equivalence, there is only a finite number of such neighbourhoods. We give a finite combinatorial presentation of the global dynamics on any such neighbourhood. A key step is the construction of a unique model of the germ of X along K; this model is, roughly speaking, the simplest three-dimensional manifold and the simplest Smale flow exhibitin…

Transitive partially hyperbolic diffeomorphisms on 3-manifolds

Abstract 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 T 2 , perturbations of the time one map of a transitive Anosov flow, and certain derived from Anosov diffeomorphisms of the torus T 3 . In this work we characterize the two first types by a local hypothesis associated to one closed periodic curve.

Hyperbolicity as an obstruction to smoothability for one-dimensional actions

Ghys and Sergiescu proved in the $80$s that Thompson's group $T$, and hence $F$, admits actions by $C^{\infty}$ diffeomorphisms of the circle . They proved that the standard actions of these groups are topologically conjugate to a group of $C^\infty$ diffeomorphisms. Monod defined a family of groups of piecewise projective homeomorphisms, and Lodha-Moore defined finitely presentable groups of piecewise projective homeomorphisms. These groups are of particular interest because they are nonamenable and contain no free subgroup. In contrast to the result of Ghys-Sergiescu, we prove that the groups of Monod and Lodha-Moore are not topologically conjugate to a group of $C^1$ diffeomorphisms. Fur…

Flexible periodic points

We define the notion of ${\it\varepsilon}$-flexible periodic point: it is a periodic point with stable index equal to two whose dynamics restricted to the stable direction admits ${\it\varepsilon}$-perturbations both to a homothety and a saddle having an eigenvalue equal to one. We show that an ${\it\varepsilon}$-perturbation to an ${\it\varepsilon}$-flexible point allows us to change it to 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 points is a general phenomenon among systems with a robustly non-hyperbolic two-dimensional center-stable bundle.

Topological classification of gradient-like diffeomorphisms on 3-manifolds

Abstract 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.

Champs de vecteurs analytiques commutants, en dimension 3 ou 4: existence de zeros communs

One proves the existence of a common zero for any two ℝ-analytic commuting vector fields on a 4-dimensional manifold with not zero Euler characteristic. A local version of this result remains true on 3-manifolds.

Aperiodic chain recurrence classes of $C^1$-generic diffeomorphisms

We consider the space of $C^1$-diffeomorphims equipped with the $C^1$-topology on a three dimensional closed manifold. It is known that there are open sets in which $C^1$-generic diffeomorphisms display uncountably many chain recurrences classes, while only countably many of them may contain periodic orbits. The classes without periodic orbits, called aperiodic classes, are the main subject of this paper. The aim of the paper is to show that aperiodic classes of $C^1$-generic diffeomorphisms can exhibit a variety of topological properties. More specifically, there are $C^1$-generic diffeomorphisms with (1) minimal expansive aperiodic classes, (2) minimal but non-uniquely ergodic aperiodic c…

Existence de points fixes enlacés à une orbite périodique d'un homéomorphisme du plan

Let f be an orientation-preserving homeomorphism of the plane such that f-Id is contracting. Under these hypotheses, we establish the existence, for every periodic orbit, of a fixed point which has nonzero linking number with this periodic orbit.