Orders of accumulation of entropy

Volume 216 / 2012

David Burguet, Kevin McGoff Fundamenta Mathematicae 216 (2012), 1-53 MSC: Primary 37A35; Secondary 46A55. DOI: 10.4064/fm216-1-1


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.


  • David BurguetCMLA-ENS Cachan
    61 avenue du président Wilson
    94235 Cachan Cedex, France
  • Kevin McGoffMathematics Department
    Duke University, Box 90320
    Durham, NC 27708-0320, U.S.A.

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