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.
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arXiv subjects
Sylvain Crovisier
Publications and source records attributed to Sylvain Crovisier.
Iterated Functions Systems, Blenders and Parablenders
We recast the notion of parablender introduced in [Berger2015] as a parametric IFS. This is done using the concept of open covering property and looking to parametric IFS as systems acting on jets.
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A dynamical decomposition of the torus into pseudo-circles
We build an irrational pseudo-rotation of the 2-torus which is semiconjugate to an irrational rotation of the circle in such a way that all the fibres of the semi-conjugacy are pseudo-circles. The proof uses the well-known `fast-approximation method' introduced by Anosov and Katok.
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Hyperbolicity versus non-hyperbolic ergodic measures inside homoclinic classes
We prove that, for $C^1$-generic diffeomorphisms, if a homoclinic class is not hyperbolic, then there is a non-hyperbolic ergodic measure supported on it. This proves a conjecture by Díaz and Gorodetski [28]. We also discuss the conjectured existence of periodic points with different stable dimension in the class.
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Dynamics of $C^1$-diffeomorphisms: global description and prospects for classification
We are interested in finding a dense part of the space of $C^1$-diffeomorphisms which decomposes into open subsets corresponding to different dynamical behaviors: we discuss results and questions in this direction. In particular we present recent results towards a conjecture by J. Palis: any system can be approximated either by one which is hyperbolic (and whose dynamics is well understood) or by one which exhibits a homoclinic bifurcation (a simple local configuration involving one or two periodic orbits).
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On the density of singular hyperbolic three-dimensional vector fields: a conjecture of Palis
In this note we announce a result for vector fields on three-dimensional manifolds: those who are singular hyperbolic or exhibit a homoclinic tangency form a dense subset of the space of $C^1$-vector fields. This answers a conjecture by Palis. The argument uses an extension for local fibered flows of Mañé and Pujals-Sambarino's theorems about the uniform contraction of one-dimensional dominated bundles. Sur la densité de l'hyperbolicité singulière pour les champs de vecteurs en dimension trois : une conjecture de Palis Dans cette note, nous annonçons un résultat portant sur les champs de vecteurs des variétés de dimension $3$ : ceux qui vérifient l'hyperbolicité singulière ou qui possèdent une tangence homocline forment un sous-ensemble dense de l'espace des champs de vecteurs $C^1$. Ceci répond à une conjecture de Palis. La démonstration utilise une généralisation pour les flots fibrés locaux des théorèmes de Mañé et Pujals-Sambarino traitant de la contraction uniforme de fibrés unidimensionnels dominés.
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Center manifolds for partially hyperbolic set without strong unstable connections
We consider compact sets which are invariant and partially hyperbolic under the dynamics of a diffeomorphism of a manifold. We prove that such a set K is contained in a locally invariant center submanifold if and only if each strong stable and strong unstable leaf intersects K at exactly one point.
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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.
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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.
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Partial Hyperbolicity and Homoclinic Tangencies
We show that any diffeomorphism of a compact manifold can be C1 approximated by diffeomorphisms exhibiting a homoclinic tangency or by diffeomorphisms having a partial hyperbolic structure.
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Perturbations of C1-diffeomorphisms and dynamics of generic conservative diffeomorphisms of surface
In the first part of this text we give a survey of the properties satisfied by the C1-generic conservative diffeomorphisms of compact surfaces. The main result that we will discuss is that a C1-generic conservative diffeomorphism of a connected compact surface is transitive. It is obtain as a consequence of a connecting lemma for pseudo-orbits. In the last parts we expose some recent developments of the C1-perturbation technics and the proof of this connecting lemma. We are not aimed to deal with technicalities nor to give the finest available versions of these results. Besides this theory exists also in higher dimension and in the non-conservative setting, we restricted the scope of this presentation to the conservative case on surfaces, since it offers some simplifications which allow to explain in an easier way the main ideas of the subject.
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Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms
We prove that any diffeomorphism of a compact manifold can be approximated in topology C1 by another diffeomorphism exhibiting a homoclinic bifurcation (a homoclinic tangency or a heterodimensional cycle) or by one which is essentially hyperbolic (it has a finite number of transitive hyperbolic attractors with open and dense basin of attraction).
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Birth of homoclinic intersections: a model for the central dynamics of partially hyperbolic systems
We prove a conjecture of J. Palis: any diffeomorphism of a compact manifold can be C1-approximated by a Morse-Smale diffeomorphism or by a diffeomorphism having a transverse homoclinic intersection. ----- Cr'eation d'intersection homoclines : un mod`ele pour la dynamique centrale des syst`emes partiellement hyperboliques. Nous montrons une conjecture de J. Palis : tout diff'eomorphisme d'une vari'et'e compacte peut ^etre approch'e en topologie C1 par un diff'eomorphisme Morse-Smale ou par un diff'eomorphisme ayant une intersection homocline transverse.
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Perturbation de la dynamique de difféomorphismes en topologie C^1 / Perturbation of the dynamics of diffeomorphisms in the C^1-topology
Les travaux présentés dans ce mémoire portent sur la dynamique de difféomorphismes de variétés compactes. Pour l'étude des propriétés génériques ou pour la construction d'exemples, il est souvent utile de savoir perturber un système. Ceci soulève généralement des problèmes délicats : une modification locale de la dynamique peut engendrer un changement brutal du comportement des orbites. En topologie C^1, nous proposons diverses techniques permettant de perturber tout en contrôlant la dynamique : mise en transversalité, connexion d'orbites, perturbation de la dynamique tangente, réalisation d'extensions... Nous en tirons diverses applications à la description de la dynamique des difféomorphismes C^1-génériques. This memoir deals with the dynamics of diffeomorphisms of compact manifolds. For the study of generic properties or for the construction of examples, it is often useful to be able to perturb a system. This generally leads to delicate problems: a local modification of the dynamic may cause a radical change in the behavior of the orbits. For the C^1 topology, we propose various techniques which allow to perturb while controlling the dynamic: setting in transversal position, connection of orbits, perturbation of the tangent dynamics,... We derive various applications to the description of the C^1-generic diffeomorphisms.
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Partial hyperbolicity far from homoclinic bifurcations
We prove that any diffeomorphism of a compact manifold can be C^1-approximated by a diffeomorphism which exhibits a homoclinic bifurcation (a homoclinic tangency or a heterodimensional cycle) or by a diffeomorphism which is partially hyperbolic (its chain-recurrent set splits into partially hyperbolic pieces whose centre bundles have dimensions less or equal to two). We also study in a more systematic way the central models introduced in arXiv:math/0605387.
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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.
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Realisation of measured dynamics as uniquely ergodic minimal homeomorphisms on manifolds
We prove that the family of measured dynamical systems which can be realised as uniquely ergodic minimal homeomorphisms on a given manifold (of dimension at least two) is stable under measured extension. As a corollary, any ergodic system with an irrational eigenvalue is isomorphic to a uniquely ergodic minimal homeomorphism on the two-torus. The proof uses the following improvement of Weiss relative version of Jewett-Krieger theorem: any extension between two ergodic systems is isomorphic to a skew-product on Cantor sets.
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The C1 generic diffeomorphism has trivial centralizer
Answering a question of Smale, we prove that the space of C1 diffeomorphisms of a compact manifold contains a residual subset of diffeomorphisms whose centralizers are trivial.
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