Ahlfors regularity and fractal dimension of Smale spaces
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
Cite
@article{arxiv.2004.07367,
title = {Ahlfors regularity and fractal dimension of Smale spaces},
author = {D. M. Gerontogiannis},
journal= {arXiv preprint arXiv:2004.07367},
year = {2022}
}
Comments
47 pages, version to appear in Ergodic Theory and Dynamical Systems. Contains an extended literature review on Ahlfors regularity in hyperbolic dynamical systems, more examples and counterexamples for understanding the main results. Bowen's equation has been removed to reduce complexity, as it was not required for deriving the main results and since it will be studied in a future project