Pseudo–Anosov mapping classes from pure mapping classes

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. Throughout this study, we also give various ways of constructing pseudo–Anosov mapping classes. Furthermore, we are able to capture the stable lengths of all pseudo–Anosov mapping classes constructed by our methods.