Survey on constructions of pseudo-Anosov surface homeomorphisms

Abstract

Pseudo-Anosov homeomorphisms are a special type in the mapping class group of a surface. Among all mapping classes, pseudo-Anosov homeomorphisms play the most important role for understanding the mapping class group. However, it is difficult to imagine an easy example of pseudo-Anosov homeomorphism. In this paper, we introduce two constructions of pseudo-Anosov homeomorphisms. Penner and Thurston found sufficient conditions to obtain pseudo-Anosov homeomorphisms using Dehn twists with curves satisfying space filling condition. Furthermore, we will focus on some meaningful properties of the constructions. There is a natural question as follows: How many pseudo-Anosov homeomorphisms arise from their constructions? For this reason, using properties of the constructions, we will show the existences of pseudo-Anosov homeomorphisms not arising from their constructions.