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Posts Tagged ‘smooth dynamics’

With Sylvain Crovisier and Todd Fisher, we have just finished the paper Local perturbations of conservative C1-diffeomorphisms. It is now available on arxiv.

While studying the entropy of diffeomorphisms in this paper, we needed a number of lemmas for perturbing the dynamics of periodic orbits  with support contained in a small neighborhood and preserving given homoclinic relations. Such results were known in the dissipative setting (though some are quite recent, see Nicolas Gourmelon‘s work).  However,  the conservative (i.e., volume-preserving or symplectic) versions were often weaker or even lacking. The new preprint fills this gap.

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Viviane Baladi a donné un exposé au séminaire de topologie et dynamique à Orsay sur le problème suivant:

Le flot du billard de Sinaï est-il exponentiellement mélangeant?

Plus précisément, on considère un point matériel se déplaçant à vitesse constante dans \mathbb T^2\setminus O, où O est une union finie d’obstacles strictement convexes, lisses, sur le bord desquels s’effectue une collision élastique. On suppose également que le temps entre deux collisions successives est borné. La mesure de Liouville (ou plutôt la mesure m induite sur une hypersurface d’énergie constante, i.e., de vitesse fixée) est invariante pour ce flot \phi_t, en fait ergodique et mélangeante d’après Sinaï. Si f,g sont deux fonctions L^2(m) sur l’espace des phases, alors \left|m(f\cdot g\circ\phi_t) - m(f)m(g)\right|\to 0 quand t\to\infty. Le problème est d’obtenir une borne de la forme:

\left|m(f\cdot g\circ\phi_t) - m(f)m(g)\right| \leq C(f,g)\exp-ct

avec C(f,g)<\infty et c>0.

Ce genre d’estimées est bien connu depuis les années 1970 pour les difféomorphismes d’Anosov. Le cas des flots est nettement plus délicat. Le mélange peut ne pas avoir lieu pour des raisons triviales (ex: suspension de temps constant). Il est cependant connu dans un certain nombre de cas particulier:  Ratner pour le flot géodésique à courbure négative constante, Dolgopyat 1998 sous une condition supplémentaire ou Liverani pour les flots d’Anosov de contact.

Le billard de Sinai peut être vu comme un flot géodésique sur une variété dont la courbe négative a été “concentrée” en des singularités. Il faut donc être capable d’analyser non seulement des flots, mais des flots singuliers.

En 1998, L.-S. Young a réussi à montrer le mélange exponentiel pour l’application de premier retour définie par le billard et sa surface de collision. Il s’est avéré impossible jusqu’ici de déduire de ce résultat le mélange exponentiel pour le flot. En 2007, N. Chernov a obtenu une décroissance sous-exponentielle de la forme e^{-c\sqrt{t}}.

L’idée de Baladi et ses collaborateurs est de contourner la construction d’une extension de Young au profit d’une approche plus directe sur la base de l’étude de l’opérateur de transfert naturel. Cet opérateur est l’action de la dynamique sur les densités. L’existence de directions contractantes le long desquelles les densités deviennent de plus en plus singulières requiert l’utilisation d’espaces fonctionnels adaptés comme l’ont montré en 2002 Blank, Keller et Liverani  et de nombreux autres travaux importants (Baladi, Tsujii, Gouëzel, Liverani).

Théorème (Baladi-Liverani). Soit M^3 compacte. Soit \phi_t un flot C^2 sur un nombre fini de morceaux avec extension. Supposons:

  1. hyperbolicité uniforme définie par deux champs de cônes
  2. préservation d’une forme de contact
  3. mélange pour le volume défini par cette forme
  4. complexité sous-exponentielle de la partition des itérées du flot et de son inverse
  5. transversalité des cônes stables par rapport aux bords des morceaux

Alors il y a mélange exponentiel pour les fonctions f,g hölderiennes.

Remarques:

  1. il n’y pas d’hypothèse markovienne sur les images des morceaux.
  2. la condition de complexité est satisfaite en dimension deux d’après un résultat de Katok. En dimension supérieure la condition semble mal comprise.
  3. les singularités du billard sont plus méchantes: il n’y a pas d’extension C^2, les dérivées explosant polynomialement près des bords. C’est la seule condition qui n’est pas satisfaite par le billard de Sinaï (en dimension 2). Des travaux sont en cours avec Balint et Gouëzel.
  4. En dimension supérieure pourrait apparaître une condition de pincement liée à la nécessité d’une certaine régularité de “pseudo-feuilletages instables”.

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We consider a diffeomorphism f:M\to M where M is a compact manifold.

A topological attractor is a compact subset \Lambda\subset M which is (i) invariant: f(\Lambda)=\Lambda; (ii) chain recurrent: for any x,y\in\Lambda, any \epsilon>0, there exists a finite sequence x_0=x,x_1,\dots,x_N=y\in M such that d(f(x_i),x_{i+1})<\epsilon; and (iii) whose basin, \{x\in M:\lim_{n\to\infty} d(f^nx,\Lambda)=0\},  is a neighborhood of \Lambda. This last property can be stated as: it admits a neighborhood U such that \Lambda=\bigcap_{n\geq0} f^n(U).

One might be tempted to think that any (or almost every) diffeomorphism admits at least one attractor. This is the case on surfaces (according to Pujals-Sambarino) and in higher dimension far from homoclinic tangencies (according to Crovisier-Pujals).  However this fails in full generality.

More precisely, in a 2009 preprint, Bonatti, Li and Yang have built, on any compact manifold of dimension at least 3,  a locally C^1-generic diffeomorphism with no topological attractor (i.e., this property holds for all diffeomorphism in a subset containing a dense intersection of countably many open subsets of some fixed open subset of Diff^1(M)).

Rafael Potrie has explained to our work group (on June 21)  his construction of such an example (see the preprint here). His example is derived from Anosov. This  allows a precise control of its dynamics:

  • there is a unique minimal Milnor attractor. More precisely, it is a compact invariant set which is chain recurrent, whose basin has full Lebesgue measure and such that any invariant proper compact subset has a basin of zero Lebesgue measure.
  • each chain recurrent class besides the Milnor attractor is contained in a cycle of center-stable-leaves.

He carefully explained his construction. He starts with a hyperbolic automorphism of the 3-torus with a simple expanding eigenvalue and a pair of conjugate, non-real, contracting eigenvalues. As in Bonatti-Viana’s paper, one modifies it in a ball of some small radius \delta>0 around a fixed point q to get:

  • q becomes a saddle point with one wealky contracting eigenvalues and two expanding ones such that the area of any 2-plane is expanding. Also, its local stable curve has length larger than \delta;
  • there are thin unstable, resp., center-stable cones, invariant under f, resp., f^{-1}, and close to the unstable, resp. stable, spaces of the linear map.

According to Bonatti-Viana, there is a constant L<\infty such that, any center-stable disk of radius bigger than 2\delta and any unstable curve of length at least L intersect.

Recall that a quasi-attractor is a compact subset \Lambda satisfying (i), (ii) and the following weakening version of (iii): it admits a decreasing sequence of neighborhoods U_n such  that \Lambda = \bigcap_{n\geq0} U_n and \overline{f(U_n)} \subset  U_n. An application of Zorn’s Lemma shows that all homeomorphisms of compact sets have quasi-attractors.

Potrie shows that his diffeomorphism has a unique quasi-attractor by observing that any small piece of unstable curve contained in any U_n is eventually mapped into a long curve that must meet the large stable disk of a fixed point r away from p,q. It follows that any quasi-attractor contains r. Hence the uniqueness.

To prove that the quasi-attractor is not an attractor, Potrie observes first that it must contains q and that, generically by a result of Bonatti-Crovisier, it must coincide with the homoclinic class of q. The choice of the eigenvalues of the automorphism implies that the center-stable space of this class has no finer dominated splitting. T_xf|E^{cs}  contracts areas. By Bonatti-Diaz-Pujals, this generically implies that the class is accumulated by sources.

The proof that the Milnor attractor is minimal is done in two steps. For C^2 systems, it follows from the existence of a Sinai-Ruelle-Bowen measure.  For C^1 systems, it follows from the fact that the existence of a minimal Milnor attractor supported by the closure of the unstable set of r is a G_\delta, which is dense by the C^2 result.

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E. Militon explained to the Groupe de travail de théorie ergodique in Orsay the following result.

Let G be a group. Given S\subset G and g\in G, the length \ell_S(g) is the minimum length of a product of elements of S\cup S^{-1} equal to g (possibly \infty).

Definition. g\in G is a distortion element if there exists a finite subset S\subset G, such that \lim_{n\to\infty} \ell_S(g^n)/n = 0.

Let G=Diff^\infty_0(M) be the group of C^\infty-smooth diffeomorphisms of a compact manifold M which, moreover, are isotopic to the identity. Let $d$ be a metric on G which is compatible with the C^\infty-topology.

Theorem (Militon). If g\in G is recurrent, i.e., \liminf_{n\to\infty} d(g^n,Id)=0, then g is a distortion element.

Avila proved a similar result in the case M=\mathbb S^1 and Militon’s proof follows Avila’s.

Lemma 1. There exist two numerical sequences \epsilon_n,k_n such that for any sequence of diffeomorphisms h_n\in Diff^\infty_0(M) satisfying d(h_n,Id)<\epsilon_n there exists a finite subset S\subset G such that \ell_S(h_n)\leq k_n for all n\geq1.

This Lemma is easily seen to imply the theorem. It is deduced from the next lemma using a non-trivial result on the decomposition of a diffeomorphism close enough to the identity into a composition of a bounded number of commutators of diffeomorphisms with small supports and themselves close to the identity.

Diff^\infty_0(\mathbb R^d) denotes the set of smooth diffeomorphisms of \mathbb R^d which are compact supported and are isotopic to the identity through a path of diffeomorphisms with supports all included in a fixed compact set of $\mathbb R^d$.

Lemma 2. There exist two numerical sequences \epsilon_n,k_n such that for any pair of sequences of diffeomorphisms f_n,g_n\in Diff^\infty_0(\mathbb R^d) satisfying d(f_n,Id)<\epsilon_n, supp(f_n)\subset B(0,1) (and similar conditions on g_n), there exists a finite subset S\subset G such that \ell_S([f_n,g_n])\leq k_n for all n\geq1.

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A major part of dynamical system theory studies the iteration of diffeomorphisms under various assumptions, geometric or analytic or otherwise. It is also interesting to study them collectively. In particular, Diff^r(M), the set of C^r diffeomorphisms of some manifold M and some number r\geq1, is a group. It is for instance of interest to know when it is simple.

Todd Fisher gave a talk in Orsay about the (non)triviality of the centralizer for many elements. Recall that the centralizer of an element g of a group G is Z(g):=\{h\in G:hg=gh\}, the set of all elements that commute with g. It is also the set of all elements h that conjugate g with itself, i.e., it can be interpreted as the set of symmetries of g.

Z(g) may be very large, e.g., equal to the whole of G if this group is Abelian. It is said to be trivial if it is reduced to \{g^n:n\in\mathbb Z\} (which is always part of it). Note that such a diffeomorphism cannot be embedded in a smooth flow.

One expects that the set \mathcal T^r(M) of C^r-diffeomorphisms of M with trivial centralizer is very big (residual or even open and dense). To show this for all compact manifolds and all 1\leq r\leq\infty) was one of Smale’s questions for the the twenty-first century.

Results are known in dimension one or C^1-smoothness under hyperbolicity assumptions:

It has turned out that the above is really only generic and fails to hold on an open and dense set according to Bonatti, Crovisier, Vago and Wilkinson. On the circle, this holds only on a set with empty interior. This last result extends to tori in dimensions up to 4 according to a work in progress of Bakker and Fisher.

Hyperbolicity also allows some results:

  • a generic C^1-axiom A diffeomorphisms with no cycles has trivial centralizer (Tugawa 1978);
  • there is an open and dense set of surface C^\infty axiom-A diffeomorphisms with no cycles having a trivial C^1-centralizer (Palis-Yoccoz 1989). This even holds for arbitrary dimensions if one assumes the existence of a sink or a source.

On surfaces, Fisher has extended the last Palis-Yoccoz result to intermediate smoothness, i.e., C^r, 2\leq r\leq \infty and axiom A with no-cycles, i.e., \Omega-stable, diffeomorphisms instead of the above ones with strong transversality, i.e., the structurally stable ones.

The proof of Palis and Yoccoz (and of its later generalizations) relies on a local theorem showing that on each basin, two commuting maps must be iterates one of the other and a global theorem that connects the basins and identify the iterates. The last is the delicate part of the proof.

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Erwan Lanneau gave a talk in Orsay about his work on surfaces of translation. Pseudo-Anosov homeomorphisms are homeomorphisms of such surfaces which are affine away from the singular set of the surface and whose differential is hyperbolic. The dilation factor is the dominant eigenvalue of that differential. It is a Perron number.

The minimum of the dilation factor for given genus is known for geni 1 and 2 only (the techniques behind could be extended to genus 3 but not farther). One also knows that c_1/g\leq \log \delta(g)\leq c_2/g.

The main result of the talk is that the above does not hold when restricted to a given type of translation surfaces. More precisely, the moduli space of translation surfaces of given genus splits into connected components (at most three), one of them corresponding to hyperellipticity and the following holds:

Theorem (Boissy-Lanneau) Let \Phi be a pseudo-Anosov on a hyperelliptic translation surface of genus g admitting an involution with 2g+2 fixed points). Assume that \Phi has a unique singularity. Then its dilation is strictly greater than \sqrt{2} (but approach this value as g\to\infty).

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Most of topological dynamics studies systems of the form T:X\to X where T is a continuous self-map and X is a compact metric space. One approach is to “reduce” such systems to symbolic dynamical system, i.e., \sigma:S\to X where S is a closed subset of \{1,\dots,d\}^{\mathbb Z} and \sigma((x_n)_{n\in\mathbb Z})=(x_{n+1})_{n\in\mathbb Z} such that \sigma(S)=S.

J. Auslander asked about the obstructions for a topological system T:X\to X to have a symbolic extension, i.e., a symbolic system \sigma:S\to S and a continuous surjection \pi:S\to X commuting with the dynamics: \pi\circ\sigma =T\circ\pi. There is an obvious one: a symbolic system (and therefore its topological factors) has finite topological entropy. Is there any other?

M. Boyle showed that this was indeed the case. With D. Fiebig and U. Fiebig, he showed that asymptotically h-expansive systems (including C^\infty self-maps of compact manifolds by a result of mine based on Yomdin’s theory) always have a “nice” symbolic extension. T. Downarowicz and S. Newhouse showed that generic C^1 map have no symbolic extension whatever, leaving open the question of diffeomorphisms with finite smoothness.

T. Downarowicz and A. Maas showed that C^r interval maps also always have symbolic extensions for 1<r<\infty.

David BURGUET has finally proved the same for arbitrary C^2 surface diffeomorphisms, see his preprint here.

Behind these works there is a rich and beautiful topological/ergodic/functional-analytical theory of entropy (called the entropy structure by T. Downarowicz) which does yet have the audience it deserves, in my opinion.

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