**Title:** Decay of correlations for the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations

**Speaker:** Alexander Bufetov

**Speaker Info:** University of Chicago

**Brief Description:**

**Special Note**:

**Abstract:**

An interval exchange transformation is a piecewise isometry of an interval, obtained by cutting the interval into a finite number of subintervals, and then rearranging these subintervals according to a given permutation.Such transformations naturally arise as first return maps of measured foliations on compact surfaces. Interval exchanges exhibit both deterministic and chaotic properties, and we are still very far from a complete understanding of their dynamical behaviour.

One of the main tools in the study of interval exchanges is renormalization: the first return map of an interval exchange on a smaller subinterval is again an interval exchange. By choosing the smaller subinterval appropriately, one endows the space of interval exchange transformations with a measure-preserving dynamical system, called the Rauzy-Veech-Zorich induction map. The dynamical behaviour of an interval exchange is then encoded by the behaviour of its orbit under the induction map.

The main result of the talk is a stretched-exponential bound on the speed of mixing for the Rauzy-Veech-Zorich induction map. The proof follows the method of Markov approximations of Bunimovich and Sinai. First, the induction map is represented as a symbolic dynamical system over a countable alphabet. Then, the map is approximated by a sequence of stationary Markov chains with growing memory. The Doeblin condition is then shown to hold for these chains, and the decay of correlations follows.

A corollary of the main result is the Central Limit Theorem for the Teichmueller flow on the moduli space of abelian differentials with prescribed singularities.

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