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 , the closure of the extreme points of M, and the relative Cantor-Bendixson rank of with respect to ex(M). We also address the optimality of these bounds.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm216-1-1,
author = {David Burguet and Kevin McGoff},
title = {Orders of accumulation of entropy},
journal = {Fundamenta Mathematicae},
volume = {219},
year = {2012},
pages = {1-53},
zbl = {1252.37008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm216-1-1}
}
David Burguet; Kevin McGoff. Orders of accumulation of entropy. Fundamenta Mathematicae, Tome 219 (2012) pp. 1-53. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm216-1-1/