| 概要 | Vershik's automorphisms (sometimes also called adic transformations) are dynamical systems that represent a shift along the asymptotic foliation of a Markov chain. We shall consider two cases: when the corresponing Markov chain is autonomous, and another when the adjacency matrices of the chain are given by a stationary process. As shown by Vershik and Livshits, an example of the first type is given by primitive substitutions; an example of the second type is given by generic interval exchange transformations (where the stationary process yielding the adjecency matrices is given by an ergodic probability measure invariant under the Rauzy--Veech--Zorich induction map).
We shall construct special suspension flows over these automorphisms, whose phase space will be shown to be isomorphic to the space of paths of a bi-infinite Markov chain. The main result of the talk is a multiplicative asymptotics for ergodic integrals of these suspenion flows. These results extend the earlier work of Anton Zorich and Giovanni Forni.
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