In 1974, Thurston proved that, up to isotopy, every automorphism of a closed
orientable surface is either periodic, reducible, or pseudo-Anosov. The latter case has
led to a rich theory with applications ranging from dynamical systems to
low-dimensional topology. Associated with every pseudo-Anosov map is a real number
, 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. 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
is of the
form
,
where
is the stretch factor of a pseudo-Anosov map arising from Thurston’s construction.
Keywords
topology, pseudo-Anosov, Salem number, stretch factor,
Thurston's construction, mapping class group
