On the Measure with Maximal Entropy for the Teichmüller Flow on the Moduli Space of Abelian Differentials

The Teichmüller flow $g_t$ on the moduli space of Abelian differentials with zeros of given orders on a Riemann surface of a given genus is considered. This flow is known to preserve a finite absolutely continuous measure and is ergodic on every connected component $\mathcal H$ of the moduli space. The main result of the paper is that $\mu/\mu(\mathcal H)$ is the unique measure with maximal entropy for the restriction of $g_t$ to $\mathcal H$. The proof is based on the symbolic representation of $g_t$.