Constructing pseudo-Anosovs from expanding interval maps

TL;DR

This paper explores the construction of pseudo-Anosov homeomorphisms from expanding interval maps, classifies when this yields pseudo-Anosovs, and produces examples with specific properties for surfaces of any genus.

Contribution

It links Thurston's observed phenomenon to generalized pseudo-Anosovs and classifies conditions for successful construction, providing explicit examples with Salem number dilatations.

Findings

Classified conditions under which the construction yields pseudo-Anosov maps.

Produced pseudo-Anosovs on surfaces of all genera with Salem number dilatations.

Connected Thurston's phenomenon to a subclass of generalized pseudo-Anosovs.

Abstract

We investigate a phenomenon observed by W. Thurston wherein one constructs a pseudo-Anosov homeomorphism on the limit set of a certain lift of a piecewise-linear expanding interval map. We reconcile this construction with a special subclass of generalized pseudo-Anosovs, first defined by de Carvalho. From there we classify the circumstances under which this construction produces a pseudo-Anosov. As an application, we produce for each $g \geq 1$ a pseudo-Anosov $\phi_g$ on the surface of genus $g$ that preserves an algebraically primitive translation structure and whose dilatation $\lambda_g$ is a Salem number.