arXiv subjects

Christian Bonatti

Publications and source records attributed to Christian Bonatti.

At least 37 records · Page 2Linked to original sources

arXiv2018

Star fows and multisingular hyperbolicity

A vector field X is called a star flow if every periodic orbit, of any vector field C1-close to X, is hyperbolic. It is known that the chain recurrence classes of a generic star flow X on a 3 or 4 manifold are either hyperbolic or singular hyperbolic (see [MPP] for 3-manifolds and [GLW] on 4-manifolds). As it is defined, the notion of singular hyperbolicity forces the singularities in the same class to have the same index. However, in higher dimension (i.e $\geq 5$) \cite{BdL} shows that singularities of different indices may be robustly in the same chain recurrence class of a star flow. Therefore the usual notion of singular hyperbolicity is not enough for characterizing the star flows. We present a form of hyperbolicity (called multi-singular hyperbolic) which makes compatible the hyperbolic structure of regular orbits together with the one of singularities even if they have different indices. We show that multisingular hyperbolicity implies that the flow is star, and conversely, there is a C1-open and dense subset of the an open set of star flows which are multisingular hyperbolic. More generally, for most of the hyperbolic structures (dominated splitting, partial hyperbolicity etc...) well defined on regular orbits, we propose a way for generalizing it to a compact set containing singular points.

math.DS↗

arXiv2018

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 $μ$ which has positive entropy and is such that for a continuous function $ϕ\colon X\to\mathbb{R}$ the integral $\intϕ\,dμ$ vanishes. The criterion is an extended version of the control at any scale with a long and sparse tail technique coming from the previous works.

math.DS↗

arXiv2018

Projective cocycles over SL(2,R) actions: measures invariant under the upper triangular group

We consider the action of $SL(2,\mathbb{R})$ on a vector bundle $\mathbf{H}$ preserving an ergodic probability measure $ν$ on the base $X$. Under an irreducibility assumption on this action, we prove that if $\hatν$ is any lift of $ν$ to a probability measure on the projectivized bunde $\mathbb{P}(\mathbf{H})$ that is invariant under the upper triangular subgroup, then $\hat ν$ is supported in the projectivization $\mathbb{P}(\mathbf{E}_1)$ of the top Lyapunov subspace of the positive diagonal semigroup. We derive two applications. First, the Lyapunov exponents for the Kontsevich-Zorich cocycle depend continuously on affine measures, answering a question in [MMY]. Second, if $\mathbb{P}(\mathbf{V})$ is an irreducible, flat projective bundle over a compact hyperbolic surface $Σ$, with hyperbolic foliation $\mathcal{F}$ tangent to the flat connection, then the foliated horocycle flow on $T^1\mathcal{F}$ is uniquely ergodic if the top Lyapunov exponent of the foliated geodesic flow is simple. This generalizes results in [BG] to arbitrary dimension.

math.DS↗

arXiv2018

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. Furthermore, we show that the group of Lodha-Moore has no nonabelian $C^1$ action on the interval. We also show that many Monod's groups $H(A)$, for instance when $A$ is such that $\mathsf{PSL}(2,A)$ contains a rational homothety $x\mapsto \tfrac{p}{q}x$, do not admit a $C^1$ action on the interval. The obstruction comes from the existence of hyperbolic fixed points for $C^1$ actions. With slightly different techniques, we also show that some groups of piecewise affine homeomorphisms of the interval or the circle are not smoothable.

math.GR↗

arXiv2017

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.

math.DS↗

arXiv2017

Dominated Pesin theory: convex sum of hyperbolic measures

In the uniformly hyperbolic setting it is well known that the set of all measures supported on periodic orbits is dense in the convex space of all invariant measures. In this paper we consider the converse question, in the non-uniformly hyperbolic setting: assuming that some ergodic measure converges to a convex combination of hyperbolic ergodic measures, what can we deduce about the initial measures? To every hyperbolic measure $μ$ whose stable/unstable Oseledets splitting is dominated we associate canonically a unique class $H(μ)$ of periodic orbits for the homoclinic relation, called its \emph{intersection class}. In a dominated setting, we prove that a measure for which almost every measure in its ergodic decomposition is hyperbolic with the same index such as the dominated splitting is accumulated by ergodic measures if, and only if, almost all such ergodic measures have a common intersection class. We provide examples which indicate the importance of the domination assumption.

math.DS↗

arXiv2017

Many intermingled basins in dimension 3

We construct a diffeomorphism of $\mathbb{T}^3$ admitting any finite or countable number of physical measures with intermingled basins. The examples are partially hyperbolic with splitting $T\mathbb{T}^3 = E^{cs} \oplus E^u$ and can be made volume hyperbolic and topologically mixing.

math.DS↗

arXiv2016

Existence of common zeros for commuting vector fields on $3$-manifolds

In $64$ E. Lima proved that commuting vector fields on surfaces with non-zero Euler characteristic have common zeros. Such statement is empty in dimension $3$, since all the Euler characteristics vanish. Nevertheless, \cite{Bonatti_analiticos} proposed a local version, replacing the Euler characteristic by the Poincaré-Hopf index of a vector field $X$ in a region $U$, denoted by $\operatorname{Ind}(X,U)$; he asked: \emph{Given commuting vector fields $X,Y$ and a region $U$ where $\operatorname{Ind}(X,U)\neq 0$, does $U$ contain a common zero of $X$ and $Y$?} \cite{Bonatti_analiticos} gave a positive answer in the case where $X$ and $Y$ are real analytic. In this paper, we prove the existence of common zeros for commuting $C^1$ vector fields $X$, $Y$ on a $3$-manifold, in any region $U$ such that $\operatorname{Ind}(X,U)\neq 0$, assuming that the set of collinearity of $X$ and $Y$ is contained in a smooth surface. This is a strong indication that the results in \cite{Bonatti_analiticos} should hold for $C^1$-vector fields.

math.DS↗

arXiv2016

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 transitive Anosov flow, or of the skew product of an Anosov torus diffeomorphism by rotation of the circle.

math.DS↗

arXiv2016

A criterion for zero averages and full support of ergodic measures

Consider a homeomorphism $f$ defined on a compact metric space $X$ and a continuous map $ϕ\colon X \to \mathbb{R}$. We provide an abstract criterion, called \emph{control at any scale with a long sparse tail} for a point $x\in X$ and the map $ϕ$, that guarantees that any weak$\ast$ limit measure $μ$ of the Birkhoff average of Dirac measures $\frac1n\sum_0^{n-1}δ(f^i(x))$ is such that $μ$-almost every point $y$ has a dense orbit in $X$ and the Birkhoff average of $ϕ$ 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 set of robustly transitive partially hyperbolic diffeomorphisms with one dimensional nonhyperbolic central direction. We also obtain applications for nonhyperbolic homoclinic classes.

math.DS↗

arXiv2016

What is ... a blender?

What is a blender? In six illustrated pages we define the construction of a blender and the role it plays in the study of smooth dynamical systems.

math.DS↗

arXiv2016

On the existence of non-hyperbolic ergodic measures as the limit of periodic measures

[GIKN] and [BBD1] propose two very different ways for building non hyperbolic measures, [GIKN] building such a measure as the limit of periodic measures and [BBD1] as the $ω$-limit set of a single orbit, with a uniformly vanishing Lyapunov exponent. The technique in [GIKN] was essentially used in a generic setting, as the periodic orbits were built by small perturbations. It is not known if the measures obtained by the technique in [BBD1] are accumulated by periodic measures. In this paper we use a shadowing lemma from [G]: $\bullet$for getting the periodic orbits in [GIKN] without perturbing the dynamics, $\bullet$for recovering the compact set in [BBD1] with a uniformly vanishing Lyapunov exponent by considering the limit of periodic orbits. As a consequence, we prove that there exists an open and dense subset $\mathcal{U}$ of the set of robustly transitive non-hyperbolic diffeomorphisms far from homoclinic tangencies, such that for any $f\in\mathcal{U}$, there exists a non-hyperbolic ergodic measure with full support and approximated by hyperbolic periodic measures. We also prove that there exists an open and dense subset $\mathcal{V}$ of the set of diffeomorphisms exhibiting a robust cycle, such that for any $f\in\mathcal{V}$, there exists a non-hyperbolic ergodic measure approximated by hyperbolic periodic measures.

math.DS↗

arXiv2016

Anomalous partially hyperbolic diffeomorphisms II: stably ergodic examples

We construct examples of robustly transitive and stably ergodic partially hyperbolic diffeomorphisms $f$ on compact $3$-manifolds with fundamental groups of exponential growth such that $f^n$ is not homotopic to identity for all $n>0$. These provide counterexamples to a classification conjecture of Pujals.

math.DS↗

arXiv2016

Transverse foliations on the torus $\T^2$ and partially hyperbolic diffeomorphisms on 3-manifolds

In this paper, we prove that given two $C^1$ foliations $\mathcal{F}$ and $\mathcal{G}$ on $\mathbb{T}^2$ which are transverse, there exists a non-null homotopic loop $\{Φ_t\}_{t\in[0,1]}$ in $\diff^{1}(\T^2)$ such that $Φ_t(\calF)\pitchfork \calG$ for every $t\in[0,1]$, and $Φ_0=Φ_1= Id$. As a direct consequence, we get a general process for building new partially hyperbolic diffeomorphisms on closed $3$-manifolds. \cite{BPP} built a new example of dynamically coherent non-transitive partially hyperbolic diffeomorphism on a closed $3$-manifold, the example in \cite{BPP} is obtained by composing the time $t$ map, $t>0$ large enough, of a very specific non-transitive Anosov flow by a Dehn twist along a transverse torus. Our result shows that the same construction holds starting with any non-transitive Anosov flow on an oriented $3$-manifold. Moreover, for a given transverse torus, our result explains which type of Dehn twists lead to partially hyperbolic diffeomorphisms.

math.DS↗

arXiv2015

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.

math.DS↗

arXiv2015

Foliated hyperbolicity and foliations with hyperbolic leaves

Given a lamination in a compact space and a laminated vector field $X$ which is hyperbolic when restricted to the leaves of the lamination, we distinguish a class of $X$-invariant probabilities that describe the behaviour of almost every $X$-orbit in every leaf, that we call u-Gibbs states. We apply this to the case of foliations in compact manifolds having leaves with negative curvature, using the foliated hyperbolic vector field on the unit tangent bundle to the foliation generating the leaf geodesics. When the Lyapunov exponents of such an ergodic u-Gibbs states are negative, it is an SRB-measure (having a positive Lebesgue basin of attraction). When the foliation is by hyperbolic leaves, this class of probabilities coincide with the classical harmonic measures introduced by L. Garnett. If furthermore the foliation is transversally conformal and does not admit a transverse invariant measure we show that the ergodic u-Gibbs states are finitely many, supported each in one minimal set of the foliation, have negative Lyapunov exponents and the union of their basins of attraction has full Lebesgue measure. The leaf geodesics emanating from a point have a proportion whose asymptotic statistics is described by each of these ergodic u-Gibbs states, giving rise to continuous visibility functions of the attractors. Reversing time, by considering $-X$, we obtain the existence of the same number of repellors of the foliated geodesic flow having the same harmonic measures as projections to $M$. In the case of only 1 attractor, we obtain a North to South pole dynamics.

math.DS↗

arXiv2015

Volume hyperbolicity and wildness

It is known that volume hyperbolicity (partial hyperbolicity and uniform expansion or contraction of the volume in the extremal bundles) is a necessary condition for robust transitivity or robust chain recurrence hence for tameness. In this paper, on any 3-manifold we build examples of quasi-attractors which are volume hyperbolic and wild at the same time. As a main corollary, we see that, for any closed 3-manifold $M$, the space $\mathrm{Diff}^1(M)$ admits a non-empty open set where every $C^1$-generic diffeomorphism has no attractors or repellers. The main tool of our construction is the notion of flexible periodic points introduced by the authors. For ejecting the flexible points from the quasi-attractor, we control the topology of the quasi-attractor using the notion of partially hyperbolic filtrating Markov partition, which we introduce in this paper.

math.DS↗

arXiv2015

A spectral-like decomposition for transitive Anosov flows in dimension three

Given a (transitive or non-transitive) Anosov vector field $X$ on a closed three-dimensional manifold $M$, one may try to decompose $(M,X)$ by cutting $M$ along two-tori transverse to $X$. We prove that one can find a finite collection $\{T_1,\dots,T_n\}$ of pairwise disjoint, pairwise non-parallel incompressible tori transverse to $X$, such that the maximal invariant sets $Λ_1,\dots,Λ_m$ of the connected components $V_1,\dots,V_m$ of $M-(T_1\cup\dots\cup T_n)$ satisfy the following properties: 1, each $Λ_i$ is a compact invariant locally maximal transitive set for $X$, 2, the collection $\{Λ_1,\dots,Λ_m\}$ is canonically attached to the pair $(M,X)$ (i.e., it can be defined independently of the collection of tori $\{T_1,\dots,T_n\}$), 3, the $Λ_i$'s are the smallest possible: for every (possibly infinite) collection $\{S_i\}_{i\in I}$ of tori transverse to $X$, the $Λ_i$'s are contained in the maximal invariant set of $M-\cup_i S_i$. To a certain extent, the sets $Λ_1,\dots,Λ_m$ are analogs (for Anosov vector field in dimension 3) of the basic pieces which appear in the spectral decomposition of a non-transitive axiom A vector field. Then we discuss the uniqueness of such a decomposition: we prove that the pieces of the decomposition $V_1,\dots,V_m$, equipped with the restriction of the Anosov vector field $X$, are "almost unique up to topological equivalence".

math.DS↗