Entropy vs volume for pseudo-Anosov maps

TL;DR

This paper explores the relationship between entropy of pseudo-Anosov maps and the volume of their mapping tori, presenting theoretical insights, explicit bounds, and experimental observations that challenge existing inequalities.

Contribution

It introduces a family of pseudo-Anosov maps that violate known inequalities in unbounded geometry settings and provides explicit bounds for punctured tori.

Findings

Constructed pseudo-Anosov maps violating inequalities

Provided explicit volume bounds for punctured tori

Presented experimental observations on entropy-volume relationships

Abstract

We will discuss theoretical and experimental results concerning comparison of entropy of pseudo-Anosov maps and volume of their mapping tori. Recent study of Weil-Petersson geometry of the Teichm\"uller space tells us that they admit linear inequalities for both sides under some bounded geometry condition. We construct a family of pseudo-Anosov maps which violates one side of inequalities under unbounded geometry setting, present an explicit bounding constant for a punctured torus, and provide several observations based on experiments.