THE CENTRAL LIMIT THEOREM FOR THE TEICHMUELLER FLOW  ON
THE MODULI SPACE OF ABELIAN DIFFERENTIALS.


Alexander I. Bufetov,
The 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. Such transformations naturally arise
as first return maps of measured foliations on compact surfaces.


By assigning to a given interval exchange its first return map on a
suitably chosen subinterval, one endows the space of interval
exchanges with a renormalization dynamical system, called the
Rauzy-Veech-Zorich induction map.
The induction map admits a natural symbolic coding over a countable alphabet.

The Teichmueller flow over the moduli space of abelian differentials can be represented,
after passing to a finite cover, as the suspension flow over the natural extension of
the induction map. This representation yields a convenient symbolic coding
for the flow, similar to the coding of geodesics on the modular surface by continued fractions.

The main result of the talk is the Central Limit Theorem for the Teichmueller flow on
the moduli space of abelian differentials. The proof follows the pattern introduced
by Sinai and Ratner. First, the speed of mixing is estimated for the induction
map, and, then, the Central Limit Theorem is derived for the suspension flow using
a recent Theorem of Melbourne and Torok.

The entropy of the Teichmueller flow with respect to the smooth invariant measure has
been computed by Veech in 1986. B.M. Gurevich and the speaker have shown that the smooth measure is the
unique measure of maximal entropy for the Teichmueller flow on the moduli space of abelian differentials.