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Ahlfors regularity and fractal dimension of Smale spaces

Part of: Dynamical systems with hyperbolic behavior Topological dynamics

Published online by Cambridge University Press:  29 April 2021

DIMITRIS MICHAIL GERONTOGIANNIS
DIMITRIS MICHAIL GERONTOGIANNIS*
Affiliation:
School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow, G12 8QQ, UK
*
e-mail: [email protected]
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Abstract

We prove that, up to topological conjugacy, every Smale space admits an Ahlfors regular Bowen measure. Bowen’s construction of Markov partitions implies that Smale spaces are factors of topological Markov chains. The latter are equipped with Parry’s measure, which is Ahlfors regular. By extending Bowen’s construction, we create a tool for transferring the Ahlfors regularity of the Parry measure down to the Bowen measure of the Smale space. An essential part of our method uses a refined notion of approximation graphs over compact metric spaces. Moreover, we obtain new estimates for the Hausdorff, box-counting and Assouad dimensions of a large class of Smale spaces.

Keywords

Ahlfors regularfractal dimensionhyperbolicityMarkov partition

MSC classification

Primary: 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
Secondary: 37B10: Symbolic dynamics
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 7 , July 2022 , pp. 2281 - 2332
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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