Orders of accumulation of entropy

David Burguet ¹   ; Kevin McGoff ²

¹ CMLA-ENS Cachan 61 avenue du président Wilson 94235 Cachan Cedex, France
² Mathematics Department Duke University, Box 90320 Durham, NC 27708-0320, U.S.A.

Fundamenta Mathematicae, Tome 216 (2012) no. 1, pp. 1-53

Résumé

For a continuous map $T$ of a compact metrizable space $X$ with finite topological entropy, the order of accumulation of entropy of $T$ is a countable ordinal that arises in the context of entropy structures and symbolic extensions. We show that every countable ordinal is realized as the order of accumulation of some dynamical system. Our proof relies on functional analysis of metrizable Choquet simplices and a realization theorem of Downarowicz and Serafin. Further, if $M$ is a metrizable Choquet simplex, we bound the ordinals that appear as the order of accumulation of entropy of a dynamical system whose simplex of invariant measures is affinely homeomorphic to $M$. These bounds are given in terms of the Cantor–Bendixson rank of $\def\ex{\mathop{\rm ex}}\overline{\ex(M)}$, the closure of the extreme points of $M$, and the relative Cantor–Bendixson rank of $\def\ex{\mathop{\rm ex}}\overline{\ex(M)}$ with respect to $\def\ex{\mathop{\rm ex}}\ex(M)$. We also address the optimality of these bounds.

Détail

DOI : 10.4064/fm216-1-1

Keywords: continuous map compact metrizable space finite topological entropy order accumulation entropy countable ordinal arises context entropy structures symbolic extensions every countable ordinal realized order accumulation dynamical system proof relies functional analysis metrizable choquet simplices realization theorem downarowicz serafin further metrizable choquet simplex bound ordinals appear order accumulation entropy dynamical system whose simplex invariant measures affinely homeomorphic these bounds given terms cantor bendixson rank def mathop overline closure extreme points relative cantor bendixson rank def mathop overline respect def mathop address optimality these bounds