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

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