Pseudo-Anosov mapping classes from pure mapping classes

TL;DR

This paper investigates how products of a fixed mapping class with powers of pure mapping classes often produce pseudo-Anosov classes, providing explicit constants and analyzing their stable lengths.

Contribution

It introduces a method to generate pseudo-Anosov mapping classes from pure mapping classes with explicit bounds and length analysis.

Findings

Almost all pure mapping classes produce pseudo-Anosov classes when raised to sufficiently large powers.

An explicit constant depending on the surface determines the threshold for pseudo-Anosov behavior.

Stable lengths of the resulting pseudo-Anosov classes are explicitly characterized.

Abstract

We study types of mapping classes which arise as a product of a given mapping class and powers of certain pure mapping classes. We derive an explicit constant depending only on a surface such that almost all above pure mapping classes give rise to pseudo-Anosov type whenever their powers are larger than the constant. Furthermore, the stable lengths of pseudo-Anosov mapping classes obtained by this method are directly captured from the construction.