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
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.Date: Tuesday, January 31, 2006
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.