Topological entropy of pseudo-Anosov maps from a typical Thurston's construction
Abstract
In this paper, we develop a way to extract information about a random walk associated with a typical Thurston's construction. We first observe that a typical Thurston's construction entails a free group of rank 2. We also present a proof of the spectral theorem for random walks associated with Thurston's construction that have finite second moment with respect to the Teichm\"uller metric. Its general case was remarked by Dahmani and Horbez. Finally, under a hypothesis not involving moment conditions, we prove that random walks eventually become pseudo-Anosov. As an application, we first discuss a random analogy of Kojima and McShane's estimation of the hyperbolic volume of a mapping torus with pseudo-Anosov monodromy. As another application, we discuss non-probabilistic estimations of stretch factors from Thurston's construction and the powers for Salem numbers to become the stretch factors of pseudo-Anosovs from Thurston's construction.
Keywords
Cite
@article{arxiv.2006.10420,
title = {Topological entropy of pseudo-Anosov maps from a typical Thurston's construction},
author = {Hyungryul Baik and Inhyeok Choi and Dongryul M. Kim},
journal= {arXiv preprint arXiv:2006.10420},
year = {2021}
}
Comments
34 pages, 6 figures. Section 2.2 was partially rewritten since Theorem A in the previous version follows from a work of Leininger. The proofs of main theorems and Section 4.2 were supplemented with more details