The stratum of random mapping classes

TL;DR

This paper studies random walks on the mapping class group, showing that almost all sample paths produce pseudo-Anosov maps with geodesics in the principal stratum, answering a question by Kapovich and Pfaff.

Contribution

It demonstrates that under certain conditions, random walks almost surely yield pseudo-Anosov maps with geodesics in the principal stratum, advancing understanding of random dynamics in the mapping class group.

Findings

Almost every infinite sample path produces pseudo-Anosov maps.

Invariant geodesics lie in the principal stratum.

Answers a previously open question by Kapovich and Pfaff.

Abstract

We consider random walks on the mapping class group whose support generates a non-elementary subgroup and contains a pseudo-Anosov map whose invariant Teichm\"uller geodesic is in the principal stratum. For such random walks, we show that mapping classes along almost every infinite sample path are eventually pseudo-Anosov, with invariant Teichm\"uller geodesics in the principal stratum. This provides an answer to a question of Kapovich and Pfaff.