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Symbolic extensions for nonuniformly entropy expanding maps

100%
Burguet D.
Colloquium Mathematicum
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2010
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tom 121
|
nr 1
129-151
EN
A nonuniformly entropy expanding map is any 𝓒¹ map defined on a compact manifold whose ergodic measures with positive entropy have only nonnegative Lyapunov exponents. We prove that a $𝓒^{r}$ nonuniformly entropy expanding map T with r > 1 has a symbolic extension and we give an explicit upper bound of the symbolic extension entropy in terms of the positive Lyapunov exponents by following the approach of T. Downarowicz and A. Maass [Invent. Math. 176 (2009)].
2
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Jumps of entropy for $C^{r}$ interval maps

100%
Burguet D.
Fundamenta Mathematicae
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2015
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tom 231
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nr 3
299-317
EN
We study the jumps of topological entropy for $C^{r}$ interval or circle maps. We prove in particular that the topological entropy is continuous at any $f ∈ C^{r}([0,1])$ with $h_{top}(f) > (log⁺||f'||_{∞})/r$. To this end we study the continuity of the entropy of the Buzzi-Hofbauer diagrams associated to $C^{r}$ interval maps.
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Orders of accumulation of entropy

63%
Burguet D., McGoff K.
Fundamenta Mathematicae
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2012
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tom 216
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nr 1
1-53
EN
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 $\overline{ex(M)}$, the closure of the extreme points of M, and the relative Cantor-Bendixson rank of $\overline{ex(M)}$ with respect to ex(M). We also address the optimality of these bounds.
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