  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.

Geometric Topology