A Transcendental Invariant of Pseudo-Anosov Maps

TL;DR

This paper introduces a new invariant for pseudo-Anosov maps on surfaces, linking Thurston norm and dilatation, and demonstrates its nontriviality and implications for minimal dilatation points.

Contribution

It defines a novel invariant $A(S,)$ for pseudo-Anosov maps and explores its properties, providing answers to questions about dilatation minimal points.

Findings

$A(S,)$ is a nontrivial invariant.

Examples show minimal dilatation points need not be rational.

The invariant connects Thurston norm and dilatation in new ways.

Abstract

For each pseudo-Anosov map $\phi$ on surface $S$, we will associate it with a $\mathbb{Q}$-submodule of $\mathbb{R}$, denoted by $A(S,\phi)$. $A(S,\phi)$ is defined by an interaction between the Thurston norm and dilatation of pseudo-Anosov maps. We will develop a few nice properties of $A(S,\phi)$ and give a few examples to show that $A(S,\phi)$ is a nontrivial invariant. These nontrivial examples give an answer to a question asked by McMullen: the minimal point of the restriction of the dilatation function on fibered face need not be a rational point.