Salem number stretch factors and totally real fields arising from Thurston's construction

TL;DR

This paper demonstrates that Salem numbers and totally real fields can be realized as stretch factors of pseudo-Anosov maps constructed via Thurston's method, linking algebraic number theory with surface dynamics.

Contribution

It establishes that every Salem number has a power as a Thurston stretch factor and every totally real field arises from such a stretch factor, expanding the understanding of algebraic units in surface automorphisms.

Findings

Every Salem number has a power that is a Thurston stretch factor.

Every totally real number field can be expressed as $Q(\lambda + rac{1}{\lambda})$ for some Thurston stretch factor.

The results connect algebraic number theory with the dynamics of surface automorphisms.

Abstract

In 1974, Thurston proved that, up to isotopy, every automorphism of closed orientable surface is either periodic, reducible, or pseudo-Anosov. The latter case has lead to a rich theory with applications ranging from dynamical systems to low dimensional topology. Associated with every pseudo-Anosov map is a real number $\lambda > 1$ known as the stretch factor. Thurston showed that every stretch factor is an algebraic unit but it is unknown exactly which units can appear as stretch factors. In this paper we show that every Salem number has a power that is the stretch factor of a pseudo-Anosov map arising from a construction due to Thurston. We also show that every totally real number field $K$ is of the form $K = \mathbb{Q}(\lambda + \lambda^{-1})$, where $\lambda$ is the stretch factor of a pseudo-Anosov map arising from Thurston's construction.