Ensembles de Julia de mesure positive
Ensembles de Julia de mesure positive et disques de Siegel des polynômes quadratiques
Xavier Buff et Arnaud Chéritat ont montré que l’ensemble de Julia de certains polynômes quadratiques est de mesure de Lebesgue positive, répondant ainsi à une question ouverte depuis Fatou et Julia. Les polynômes en question ont un point fixe indifférent irrationnel dont le nombre de rotation doit être soigneusement déterminé. On exposera les grandes lignes de la démonstration, ainsi que d’autres résultats connexes des mêmes auteurs sur la géométrie et la taille des disques de Siegel.
Ensembles de torsion nulle des applications déviant la verticale
Nous définissons la notion d’ensemble bien ordonné de torsion nulle pour les applications déviant la verticale. Contrairement aux études variationnelles de [14] et [1], nous proposons une approche topologique. On retrouve pour ces ensembles un grand nombre de propriétés des ensembles bien ordonnés décrites dans [11]. En reprenant un argument de G.Hall [7], nous montrons en particulier que pour tout nombre de rotation, il existe un ensemble bien ordonné de torsion nulle.
Entropic approximation in kinetic theory
Approximation theory in the context of probability density function turns out to go beyond the classical idea of orthogonal projection. Special tools have to be designed so as to respect the nonnegativity of the approximate function. We develop here and justify from the theoretical point of view an approximation procedure introduced by Levermore [Levermore, J. Stat. Phys. 83 (1996) 1021–1065] and based on an entropy minimization principle under moment constraints. We prove in particular a global...
Entropic approximation in kinetic theory
Approximation theory in the context of probability density function turns out to go beyond the classical idea of orthogonal projection. Special tools have to be designed so as to respect the nonnegativity of the approximate function. We develop here and justify from the theoretical point of view an approximation procedure introduced by Levermore [Levermore, J. Stat. Phys.83 (1996) 1021–1065] and based on an entropy minimization principle under moment constraints. We prove in particular...
Entropie de l'image inverse d'une application
Entropies des flots magnétiques
Entropies of the partitions of the unit interval generated by the Farey tree
Entropy and complexity of a path in sub-riemannian geometry
We characterize the geometry of a path in a sub-riemannian manifold using two metric invariants, the entropy and the complexity. The entropy of a subset of a metric space is the minimum number of balls of a given radius needed to cover . It allows one to compute the Hausdorff dimension in some cases and to bound it from above in general. We define the complexity of a path in a sub-riemannian manifold as the infimum of the lengths of all trajectories contained in an -neighborhood of the path,...
Entropy and complexity of a path in sub-Riemannian geometry
We characterize the geometry of a path in a sub-Riemannian manifold using two metric invariants, the entropy and the complexity. The entropy of a subset A of a metric space is the minimum number of balls of a given radius ε needed to cover A. It allows one to compute the Hausdorff dimension in some cases and to bound it from above in general. We define the complexity of a path in a sub-Riemannian manifold as the infimum of the lengths of all trajectories contained in an ε-neighborhood of the path,...
Entropy and growth of expanding periodic orbits for one-dimensional maps
Let f be a continuous map of the circle or the interval I into itself, piecewise , piecewise monotone with finitely many intervals of monotonicity and having positive entropy h. For any ε > 0 we prove the existence of at least periodic points of period with large derivative along the period, for some subsequence of natural numbers. For a strictly monotone map f without critical points we show the existence of at least such points.
Entropy and localization of eigenfunctions
Entropy and mixing for amenable group actions.
Entropy dimension and variational principle
Recently the notions of entropy dimension for topological and measurable dynamical systems were introduced in order to study the complexity of zero entropy systems. We exhibit a class of strictly ergodic models whose topological entropy dimensions range from zero to one and whose measure-theoretic entropy dimensions are identically zero. Hence entropy dimension does not obey the variational principle.
Entropy dimension of dynamical systems.
Entropy, homology and semialgebraic geometry
Entropy linearity and chain-recurrence
Entropy of convolutions on the circle.
Entropy of distal groups, pseudogroups, foliations and laminations
A distality property for pseudogroups and foliations is defined. Distal foliated bundles satisfying some growth conditions are shown to have zero geometric entropy in the sense of É. Ghys, R. Langevin and P. Walczak [Acta Math. 160 (1988)].
Entropy of eigenfunctions of the Laplacian in dimension 2
We study asymptotic properties of eigenfunctions of the Laplacian on compact Riemannian surfaces of Anosov type (for instance negatively curved surfaces). More precisely, we give an answer to a question of Anantharaman and Nonnenmacher [4] by proving that the Kolmogorov-Sinai entropy of a semiclassical measure for the geodesic flow is bounded from below by half of the Ruelle upper bound. (This text has been written for the proceedings of the Journées EDP (Port d’Albret-June, 7-11 2010))