##### Department of Mathematics,

University of California San Diego

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### Special Colloquium

## Alexander I. Bufetov

#### Rice University

## Existence and uniqueness of the measure of maximal entropy for the Teichmuller flow on the moduli space of abelian differentials \\ \small [joint work with B.M. Gurevich]

##### Abstract:

The moduli space of abelian differentials carries a natural Lebesgue measure class, and, by the Theorem of H.Masur and W.Veech, the Teichmueller flow on the moduli space of abelian differentials preserves a finite ergodic measure in the Lebesgue measure class. The entropy of the flow with respect to the absolutely continuous measure has been computed by Veech in 1986. The main result of this talk, obtained by B.M. Gurevich and the speaker, is that the absolutely continuous measure is the unique measure of maximal entropy for the Teichmueller flow. The first step of the proof is an observation that the absolutely continuous measure has the Margulis property of uniform expansion on unstable leaves. After that, the argument proceeds in Veech's space of zippered rectangles. The flow is represented as a symbolic flow over a countable topological Bernoulli chain and with a Hoelder roof function depending only on the future. Following the method of Gurevich, the flow is then approximated by a sequence of flows whose suspension functions depend on only one coordinate in the sequence space. For these, conditions for existence and uniqueness of the measure of maximal entropy are known by theorems of Gurevich and Savchenko. Since the roof function of our initial flow is Hoelder, the approximation is rapid enough and yields maximality of entropy for the smooth measure as well as the uniqueness of the measure of maximal entropy.

Host: Efim Zelmanov

### November 27, 2006

### 3:30 PM

### AP&M 7218

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