Posts Tagged ‘genericity’

La simulation numérique est un outil central dans l’étude pratique des systèmes dynamiques. On la modélise de la façon suivante. Soit f:X\to X une application continue sur un espace métrique compact. Une discrétisation de pas \epsilon est la donnée d’une application f_*:X_*\to X_* d’une partie finie \epsilon-dense avec d(f(x),f_*(x))<\epsilon pour tout x\in X_*. Si X=\mathbb T^d, la discrétisation uniforme d’ordre N\geq1 est définie par X_N:=N^{-1}\mathbb Z^d\cap[0,1[^d+\mathbb Z^d et f_N:X_N\to X_N est une application mesurable telle que d(f_N(x),f(x))\leq d(y,f(x)) pour tout x,y\in X_N.

Problématique: les propriétés dynamiques de f:X\to X peuvent-elle se lire sur ses discrétisations et comment?

Ce sujet ne se laisse pas attaquer directement par les techniques habituelles de la théorie des systèmes dynamiques et la plupart des questions naturelles restent ouvertes malgré quelques résultats et expériences intriguantes (voici une introduction partielle et informelle à la littérature “historique”).

Pierre-Antoine Guihéneuf a donné un séminaire à Orsay sur le cas des dynamiques génériques préservant le volume. Il a annoncé:

Théorème (Guihéneuf 2015). Il existe un ensemble G_\delta dense de difféomorphismes conservatifs f:\mathbb T^d\to\mathbb T^d de classe C^1 tels que la proportion de points périodiques pour f_N tend vers 0 quand N\to\infty.

La preuve est délicate, y compris dans le cas “jouet” d’une suite d’applications linéaires. Ce résultat suggère que le comportement des discrétisations peut se comparer à celui d’une application discrète aléatoire (qui compte typiquement \sqrt{N^d} points périodiques) et s’oppose à celui observé en régularité C^0:

Théorème (Guihéneuf 2012). Il existe un ensemble G_\delta dense d’homéomorphismes conservatifs f:\mathbb T^d\to\mathbb T^d tels que:

  1. Tout point de l’intervalle est accumulé par la suite \#\{x\in X_N:\exists k>0 f_N^k(x)=x\}/\# X_N, N\to\infty;
  2. Toute mesure de probabilité borélienne f-invariante est point d’accumulation d’une suite de mesures \mu_N\mu_N est f_N-invariante.

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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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Frédéric Le Roux has written a very lucid exposition of the Alpern genericity theorem. This theorem states that the same ergodic properties are generic in the set Auto(T^d) of volume-preserving measurable transformations and in the set Homeo(T^d) of volume-preserving  homeomorphisms.

More precisely,  endow Auto(T^d)   with the weak topology, i.e., the coarsest generated by \mu\mapsto\mu(A), for all measurable subsets A and equip Homeo(T^d)  with the uniform distance.

The theorem considers any P\subset Auto(T^d)  invariant under volume-preserving measurable isomorphisms. It states that P  is a dense G_\delta subset of Auto(T^d) if and only if P\cap Homeo(T^d) is itself a dense G_\delta subset of Homeo(T^d).

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On 2009/11/27 at IHP on the occasion of the Katok 65 conference, Patrick BERNARD presented his notion of codimension in Banach spaces with applications to Mather measures and the transversality theorem.

B will denote the ambient Banach space. A large part of the theory extends to Fréchet spaces (things get tougher once C^1 smoothness is involved).

A Lipschitz graph in B of codimension k is a subset of the graph of a Lipschitz function C \to T where B=C\oplus T and \dim(T)=k. A rectifiable set of codimension k is the image by the canonical projection B\times \mathbb R^n\to B of a Lipschitz graph of codimension k+n in B\times\mathbb R^n.  A subset of B is of codimension k if it is a countable union of rectifiable subsets of that codimension.

Theorem (Zajicek 2008). The complement of a subset of positive codimension in a Banach space is Baire generic (i.e., a dense G_\delta) and prevalent.

Codimension behaves as expected with respect to C^1 maps: it is invariant under C^1 diffeomorphisms, it decreases the Fredholm index of a C^1 map, it is preserved by preimage under submersions.

It is then strikingly straightforward to prove the following applications:

Theorem. The set of potentials with k+1 Mather measures has codimension k in C^p

Theorem. The set of maps from a m-dimensional manifold to \mathbb R^n that fail to be injective immersions has codimension n-2k

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