Le théorème d’Ornstein (1970) est un des sommets de la théorie ergodique. C’est l’aboutissement des recherches initiées par Kolmogorov sur la classification des schémas de Bernoulli et le point de départ de résultats très généraux. Downarowicz et Serafin ont publié une élégante preuve de ce théorème difficile. Rappelons l’énoncé du théorème (dans sa version la [...]
Archive for the ‘papers’ Category
Une courte preuve du théorème d’Ornstein par Downarowicz et Serafin
Posted in papers, tagged classification, entropy, ergodic theory on May 20, 2012 | Leave a Comment »
Dichotomie: exposants tous nuls / hyperbolicité et ergodicité
Posted in Dynamics, papers, tagged generic dynamics, partial hyperbolicity, smooth ergodic theory on April 10, 2012 | Leave a Comment »
Dans un travail récemment diffusé sur arxiv, Jana RODRIGUEZ HERTZ montre le théorème suivant: Théorème. Soit un difféomorphisme d’une variété compacte tridimensionnelle, de classe et préservant la mesure volume . Génériquement, vérifie l’une des deux assertions suivantes: -p.p.les trois exposants de Lyapunov sont nuls; -p.p. aucun des trois exposants n’est nul. De plus (a) est [...]
Transitivité robuste en dynamique hamiltonienne
Posted in Dynamics, papers on September 9, 2011 | Leave a Comment »
Dans un article disponible sur arxiv, Nassiri et Pujals établissent des propriétés de mélange robuste pour certains systèmes hamiltoniens notamment en donnant une version symplectique des mélangeurs de Bonatti et Diaz. Une des motivations est la conjecture de diffusion d’Arnold (voir par exemple ce texte) selon laquelle, sauf en dimension 2, une dynamique définie par [...]
Origines de la diffusion (A. Kupiainen, ICM 2010)
Posted in Dynamics, mathematical physics, papers on August 5, 2011 | Leave a Comment »
L’article, disponible sur arxiv, explique l’état de la recherche sur l’origine microscopique de la diffusion dans les systèmes (microscopiquement) conservatifs. Un exemple fondamental est celui de la loi de Fourrier qui résiste encore à une analyse rigoureuse malgré son apparente évidence. Une première approche considère le couplage d’oscillateurs. Conjecturalement, les objets physiques sont les états [...]
K. McGoff: Random Subshifts of Finite Type
Posted in Dynamics, papers, tagged entropy, random, symbolic dynamics on June 9, 2010 | Leave a Comment »
It is often interesting to study the properties of objects picked at random in some class. This may shed some light on whether some observed system has is typical in some probabilistic model or may even turn out to be an efficient way of obtaining an interesting behavior. This idea was put to spectacular use [...]
Alpern genericity theorem
Posted in Dynamics, papers, tagged Dynamics, ergodic theory, genericity, topological dynamics on January 27, 2010 | Leave a Comment »
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 of volume-preserving measurable transformations and in the set of volume-preserving homeomorphisms. More precisely, endow with the weak topology, i.e., the coarsest generated by , for all measurable [...]
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 »
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 [...]
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 [...]
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 [...]
Invisible Attractors
Posted in Dynamics, papers, tagged attractors, Dynamics, smooth ergodic theory on June 22, 2009 | Leave a Comment »
Yu. Ilyashenko and A. Negut have discovered the following dynamical phenomenon. Recall that the statistical attractor is the smallest closed set such that almost every orbit spends almost all its time arbitrarily close to . They say that an open set is -invisible if almost every orbit spends a fraction of its time less than [...]
