  Let Q be a component of a stratum of abelian or quadratic differentials on an
oriented surface of genus $g\geq 0$ with $m\geq 0$ punctures and $3g-3+m\geq
2$. We construct a subshift of finite type $(\Omega,\sigma)$ and a Borel
suspension of $(\Omega,\sigma)$ which admits a finite-to-one semi-conjugacy
into the Teichmueller flow $\Phi^t$ on Q. This is used to show that the
$\Phi^t$-invariant Lebesgue measure on Q is the unique measure of maximal
entropy.
