On pseudo-Anosov maps with small dilatations on punctured Riemann spheres

TL;DR

This paper studies pseudo-Anosov maps with minimal dilatations on punctured spheres, showing they cannot be constructed via Thurston's method and cannot be trivial when punctures are filled in, with bounds on dilatations provided.

Contribution

It demonstrates that minimal dilatation pseudo-Anosov maps on punctured spheres are not obtainable from Thurston's construction and cannot be trivial when punctures are filled.

Findings

Minimal dilatation pseudo-Anosov maps cannot be obtained from Thurston's construction.

Such maps cannot define trivial mapping classes when punctures are filled.

Bounds on dilatations are established for these maps.

Abstract

Let $S_n$ be a punctured Riemann spheres $\mathbf{S}^2\backslash \{x_1,..., x_n\}$. In this paper, we investigate pseudo-Anosov maps on $S_n$ that are isotopic to the identity on $S_n\cup \{x_n\}$ and have the smallest possible dilatations. We show that those maps cannot be obtained from Thurston's construction (that is the products of two Dehn twists). We also prove that those pseudo-Anosov maps $f$ on $S_n$ with the minimum dilatations can never define a trivial mapping class as any puncture $x_i$ of $S_n$ is filled in. The main tool is to give both lower and upper bounds estimations for dilatations $\lambda(f)$ of those pseudo-Anosov maps $f$ on $S_n$ isotopic to the identity as a puncture $x_i$ of $S_n$ is filled in.