Most of topological dynamics studies systems of the form where is a continuous self-map and is a compact metric space. One approach is to “reduce” such systems to symbolic dynamical system, i.e., where is a closed subset of and such that . J. Auslander asked about the obstructions for a topological system to have a [...]
Archive for the ‘news’ Category
C^2 surface diffeomorphisms always have a symbolic extension
Posted in Dynamics, news, papers, tagged Dynamics, entropy, entropy structure, smooth dynamics, smooth ergodic theory, symbolic extension, topological dynamics on December 10, 2009 | Leave a Comment »
Discontinuity of the topological entropy for Lozi maps
Posted in Dynamics, news, papers, tagged continuity of the entropy, Dynamics, entropy, examples, hyperbolicity, Lozi maps, piecewise affine dynamics, surface dynamics on December 10, 2009 | Leave a Comment »
I have shown that, like diffeomorphisms, piecewise affine surface homeomorphisms are approximated in entropy by horseshoes, away from their singularities. It follows in particular that their topological entropy is lower-semicontinuous: a small perturbation cannot cause a macroscopic drop in entropy. The continuity of the entropy for such maps had been an open problem for some [...]
Sylvain CROVISIER’s HDR
Posted in Dynamics, news, talks, tagged bifurcations, closing lemma, Dynamics, generic dynamics, partial hyperbolicity, smooth ergodic theory, topological dynamics on November 27, 2009 | Leave a Comment »
On November 25th, 2009, Sylvain CROVISIER defended his habilitation à diriger des recherches titled Perturbation de la dynamique de difféomorphismes en petite régularité. He first explained basic perturbation techniques: the Anosov-Katok procedure: you use more and more distorted conjugacies such that the limiting dynamics has new properties; the closing lemma of Pugh and the subsequent [...]
Lower bounds for Hausdorff dimension
Posted in Dynamics, news, papers, tagged Dynamics, entropy, ergodic theory, Hausdorff dimension on October 13, 2009 | Leave a Comment »
A classical theorem (Marstrand 1954) asserts that, given any Borel subset , the obvious inequality of the Hausdorff dimensions: is in fact an equality for almost all orthogonal projections . As is often the case it is usually very dificult to prove equality for a given projection. Preliminary description: Michael HOCHMAN and Pablo SCHMERKIN have [...]
Non-convergence des mesures de Gibbs pour T->0
Posted in Dynamics, news, papers, tagged Dynamics, maximizing measures, symbolic dynamics, thermodynamical formalism on June 17, 2009 | Leave a Comment »
Soit un système dynamique hyperbolique topologiquement transitif muni d’un potentiel continu . La mesure de Gibbs à température est par définition l’unique mesure de probabilité -invariante maximisant l’énergie libre (appelée pression en dynamique…): où est l’entropie mesurée. Il est facile de voir que tout point d’accumulation de pour maximise . Pour certains systèmes, il existe [...]
Wicked and Weird
Posted in news on March 27, 2009 | Leave a Comment »
Flavio ABDENUR and Martin ANDERSSON: Ergodic theory of generic continuous maps (seminar of F.A. at Parix-XIII). Both generic endomorphisms of manifolds of any dimension and generic homeomorphisms of manifolds of dimension greater than one exhibit highly pathological ergodic properties with respect to Lebesgue measure: they are weird in the sense that they support neither physical [...]
