Small dilatation pseudo-Anosov mapping classes and short circuits on train track automata

TL;DR

This paper surveys the minimum dilatation problem for pseudo-Anosov mapping classes and provides an explicit train track description of an infinite family with conjectural minimal dilatation for certain surfaces.

Contribution

It introduces the first explicit train track description of an infinite family of pseudo-Anosov classes with minimal dilatation conjecture for even genus surfaces.

Findings

Explicit train track description of pseudo-Anosov classes

Identification of a family with conjectural minimal dilatation

Insights into the structure of minimal dilatation mapping classes

Abstract

This note gives a brief survey of the minimum dilatation problem for pseudo-Anosov mapping classes, and the first explicit train track description of an infinite family of pseudo-Anosov mapping classes with orientable stable foliations and the conjectural minimum dilatation for closed surfaces of even genus $g \ge 2$.