Topological Entropy of Random Walks on Mapping Class Groups

TL;DR

This paper establishes that for random walks on the mapping class group of a surface, the topological entropy almost surely equals the drift, linking dynamic complexity with geometric translation distance.

Contribution

It introduces the concept of topological entropy for random walks on the mapping class group and proves its almost sure equality with the drift on Teichmüller space.

Findings

Topological entropy of random walks equals the drift almost surely.

The result connects entropy with geometric translation distance.

Provides a new perspective on the dynamics of mapping class groups.

Abstract

For any pseudo-Anosov diffeomorphism on a closed orientable surface $S$ of genus greater than one, it is known by the work of Bers and Thurston that the topological entropy agrees with the translation distance on the Teichm\"uller space with respect to the Teichm\"uller metric. In this paper, we consider random walks on the mapping class group of $S$. The drift of a random walk is defined as the translation distance of the random walk. We define the topological entropy of a random walk and prove that it almost surely agrees with the drift on the Teichm\"uller space with respect to the Teichm\"uller metric.