Take a unit interval, partition it into several subintervals and rearrange these subintervals according to a given permutation. The resulting map is called an interval exchange transformation.
For example, an interval exchange of two intervals is a circle rotation.
Take an interval exchange and consider its induced map on a
subinterval: the resulting map is again an interval exchange.
By choosing the subinterval appropriately, one obtains an exchange of the same number of intervals, and, therefore, a map of the space of interval exchange transformations into itself, called the Rauzy-Veech-Zorich induction map
(in the case of circle rotations, this construction
gives the Gauss map). The induction map preserves
an absolutely continuous invariant probability measure
(constructed by Veech and Zorich), with respect
to which the map is ergodic and its square is exact.
The main result of the talk is a stretched exponential bound on the decay of correlations for the Rauzy-Veech-Zorich induction map (in the case of the Gauss map, a theorem of Kuzmin). The proof uses the method of Markov approximations of Sinai, Bunimovich-Sinai.
A corollary is the Central Limit Theorem for the Teichmueller flow on the moduli space of abelian differentials on a compact Riemann surface.
Since the Teichmueller flow can be represented as a special flow over the natural extensionof the induction map (in the case of circle rotations, this corresponds to coding geodesics on the modular surface by continued fractions), by a Theorem of Melbourne and Torok, the decay of correlations for the induction map implies the Central Limit Theorem for the Teichmueller flow.