Brandeis University Everyperson Seminar

Special time: Wednesday January 18, 2006, 3:30 PM

tea at 3:15 PM at the Math Department lounge (Goldsmith 300)

Speaker: Alexander Bufetov (Chicago/Rice)

Title: Interval exchange transformations and Teichmüller flows

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

The talk, aimed at a general audience, would require no background.