Topological entropy of pseudo-Anosov maps from a typical Thurston's construction

TL;DR

This paper investigates the properties of random walks derived from Thurston's construction, establishing spectral theorems, and demonstrating that such walks tend to produce pseudo-Anosov maps, with applications to hyperbolic volume estimation and stretch factors.

Contribution

It introduces a spectral theorem for random walks in Thurston's construction and proves that these walks typically become pseudo-Anosov, extending prior theoretical insights.

Findings

Random walks from Thurston's construction form a free group of rank 2.

Spectral theorem established for random walks with finite second moment.

Random walks eventually produce pseudo-Anosov maps under certain conditions.

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.