ABSTRACT:

A Vershik's automorpshism (also called "adic transformation") is a dynamical system whose orbits are leaves of the stable foliation of a Markov chain. The underlying Markov chain may be assumed time-homogeneous; more generally, one can consider non-homogeneous Markov chains whose transition matrices are generated according to a stationary law. An particular case is given by interval exchange maps with a periodic Rauzy expansion, or by generic interval exchange maps corresponding to an invariant measure of the Rauzy-Veech-Zorich induction map. The main result of the talk is an asymptotic formula for ergodic sums of Vershik's automorphisms. To a Vershik's automorphism it is possible to assign a natural suspension flow over it; the asymptotic formula is given in terms of special holonomy-invariant functionals on orbits of the flow. A corollary of the main result is an asymptotic formula for ergodic sums for generic interval exchange maps or translation flows on generic flat surfaces. These results extend earlier work of Anton Zorich and are close in spirit to the work of Giovanni Forni on invariant distributions for translation flows.