Decay of correlations for the Rauzy-Veech-Zorich induction map and the central limit theorem for the Teichmuller geodesic flow

The main result of the dissertation is a stretched-exponential estimate on the decay of correlations for the Rauzy-Veech-Zorich map on the space of interval exchange transformations. A corollary of the main result is the Central Limit Theorem for the Teichmuller geodesic flow on the moduli space of abelian differentials.; The first step in the proof is a construction of a countable Markov partition and the corresponding symbolic dynamics over a countable alphabet for the induction map. This allows to find a subset of the phase space such that the Rauzy-Veech-Zorich map, induced to that subset, is uniformly expanding. The next step, motivated by the Tower Method of Lai-Sang Young, is an estimate for the asymptotics of return times into the subset. The estimate is achieved by direct calculation in Veech's space of zippered rectangles.; The proof is then completed by the method of Markov approximations of Sinai, Bunimovich-Sinai. The induction map is approximated by a sequence of Markov chains. These chains satisfy the Doeblin condition, which implies the decay of correlations for them, and, consequently, also for the Rauzy-Veech-Zorich induction map.; The Teichmuller geodesic flow on the moduli space of abelian differentials can be represented, after passing to a finite cover, as a special flow over the natural extension of the induction map. Such a representation allows to derive dynamical properties of the flow from those of the map following the method of Sinai and Ratner in their proof of the Central Limit Theorem for geodesic flows on compact negatively curved manifolds. By a recent Theorem of Melbourne and Torok, the decay of correlations for the Rauzy-Veech-Zorich map implies the Central Limit Theorem for the Teichmuller geodesic flow on the moduli space of abelian differentials.