PA mapping classes with minimum dilatation and Lanneau-Thiffeault polynomials

TL;DR

This paper investigates the minimal dilatation of pseudo-Anosov (pA) diffeomorphisms on orientable surfaces of genus g, using graph theory insights to better understand the associated polynomials and their properties.

Contribution

It introduces a novel application of a lesser-known digraph theorem to analyze the minimal dilatation and related polynomials of pA diffeomorphisms.

Findings

Insight into the structure of minimal dilatation polynomials

Connection between digraph properties and dilatation bounds

Potential new bounds for dilatation values

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 $\lambda_{min,g}$.