On pseudo-Anosov mapping classes with minimum dilatation and Lanneau-Thiffeault numbers

TL;DR

This paper investigates the minimal dilatation of pseudo-Anosov diffeomorphisms on orientable surfaces, using graph theory insights to better understand the smallest possible dilatation values.

Contribution

It introduces a novel application of a lesser-known digraph theorem to analyze minimal dilatations in pseudo-Anosov mapping classes.

Findings

Provides bounds on minimal dilatations for various genera

Connects digraph properties to pseudo-Anosov dilatations

Offers new insights into Lanneau-Thiffeault numbers

Abstract

It has been known since 1981 that if one fixes an orientable surface $S$ of genus $g$, then there is a real number $\lambda_{min,g} > 1$ that is the dilatation of a pA diffeomorphism of $S$, and every other pA diffeomorphism of $S$ has dilatation $\geq \lambda_{min,g}$. We will show how a little-known theorem about digraphs gives some insight into $\lambda_{min,g}$.