For a closed, negatively curved manifold, the geodesic flow has a unique probability measure of maximal entropy: the Bowen-Margulis measure. By a theorem of Katok-Knieper-Pollicott-Weiss, the Bowen-Margulis measure is continuous under perturbations of the Riemannian metric. Instead of the fundamental group of this manifold, we can consider an arbitrary word-hyperbolic group, so that the manifold and its metric are replaced by a proper and cobounded isometric action on a geodesic metric space. For each of these isometric actions, Furman used Patterson-Sullivan measures to construct a geodesic current analogous to the Bowen-Margulis measure, now an invariant Radon measure on the space of pairs of distinct points in the boundary at infinity. In this talk, I will explain a version of Katok-Knieper-Pollicott-Weiss's theorem in the context of word-hyperbolic groups: continuous perturbations of the isometric action give a continuous perturbation of the Bowen-Margulis current.
What is the expected number of iterates of a point needed for a plot of these iterates to approximate the attractor of the dynamical system up to a given scale delta (i.e., the orbit will have visited a delta-neighbourhood of every point in the attractor)? This question has analogues in random walks on graphs and Markov chains and can be seen as a recurrence problem. I'll present joint work with Natalia Jurga (St Andrews) where we estimate the expectation for this problem as a function of delta for some classes of interval maps using ideas from Hitting Time Statistics, permutations and inducing.
Homomorphisms are topological factors between topological dynamical systems, up to GL(d,Z) transformation. This notion extends the classical dynamical ones like factor, conjugacies and automorphisms. While the automorphism group is the centralizer of the action group in the group of self-homeomorphisms in the phase space, the isomorphism group (invertible homomorphisms) is the normalizer of the action group. In this talk we will present some recent results about some rigidity properties of homomorphisms between substitutive subshifts generated by constant-shape substitutions. Constant-shape substitutions are a multidimensional generalization of constant-length substitutions, where any letter is assigned a pattern with the same shape.
Dado un subshift X y una función continua f: X → R podemos definir dos nociones con significado físico. La primera es la de medida de Gibbs, que captura la idea de equilibrio local con el entorno. La segunda es la noción de medida de equilibrio, que captura la idea de maximizar el desorden globalmente. Un teorema de Lanford y Ruelle dice que si f es suficientemente regular y X es un subshift de tipo finito en Zd, entonces las medidas de equilibrio son automáticamente medidas de Gibbs.
En esta charla presentaremos una versión "en esteroides" de ese teorema. Reemplazaremos Zd por un grupo sófico numerable arbitrario y los subshifts de tipo finito por una clase mucho más grande (los subshifts que satisfacen la propiedad topológica de Markov). Comenzaré introduciendo todos estos conceptos y luego moveré las manos enfáticamente para convencerles de que el teorema de Lanford Ruelle es válido en éste contexto.
Trabajo en conjunto con Tom Meyerovitch.