Pseudo-Anosov maps with small stretch factors on punctured surfaces

TL;DR

This paper investigates the minimal entropy of pseudo-Anosov maps on punctured surfaces, establishing that for fixed punctures, the minimum entropy scales inversely with genus, confirming a long-standing conjecture.

Contribution

It provides a precise asymptotic behavior of the minimum entropy for a large subset of surface parameters, confirming Penner's 1991 speculation.

Findings

Minimum entropy behaves as 1/g for fixed n

Confirmed Penner's conjecture from 1991

Analyzed behavior for a large subset of (g,n)

Abstract

Consider the problem of estimating the minimum entropy of pseudo-Anosov maps on a surface of genus $g$ with $n$ punctures. We determine the behaviour of this minimum number for a certain large subset of the $(g,n)$ plane, up to a multiplicative constant. In particular it has been shown that for fixed $n$, this minimum value behaves as $\frac{1}{g}$, proving what Penner speculated in 1991.