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 [...]
Posts Tagged ‘entropy’
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 »
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 [...]
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 [...]
Ergodicity of smooth systems with product measures
Posted in Dynamics, Meetings, talks, tagged entropy, generic dynamics, smooth dynamics, smooth ergodic theory on November 27, 2009 | Leave a Comment »
Federico RODRIGUEZ-HERTZ presented at the IHP conference for Katok’s 65th birthday several results pertaining to the ergodicity of systems with some product structure. Theorem (RHRHTU 2008). Let be a C^2 diffeomorphism of a compact manifold which preserves volume. Let be a hyperbolic periodic point. Define has a transverse point . Define similarly. If and then [...]
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 [...]
Première Journée Affine par Morceaux
Posted in Meetings, tagged Dynamics, entropy, group actions, piecewise affine dynamics, symbolic dynamics, topological dynamics on February 23, 2009 | Leave a Comment »
Elle aura bien lieu le jeudi 7 mai à l’Institut Henri Poincaré, Amphithéatre Darboux: voir ici. Merci de vérifier votre inscription et de vous inscrire ou désinscrire selon le cas.
Backward and Forward with the Towel Map
Posted in musings, tagged chaos, computer simulations, Dynamics, entropy, pictures, smooth ergodic theory, towel map on February 23, 2009 | Leave a Comment »
Consider the family of symmetric towel maps (the towel terminology is due to S. Newhouse): This looks like a very natural generalization of quadratic interval maps, a step beyond the Viana maps of the form . These maps can also be understood as a coupling of two chaotic interval maps. One would like to prove [...]
