Consider a regular topological Markov chain.
In 1982 Anatoly M. Vershik (note also the 1977 work of Shunji Ito) constructed a remarkable transformation whose orbits are, precisely, equivalence classes of the tail relation on the chain.
This transformation, given by a simple combinatorial construction similar to that of an adding machine, preserves a unique invariant probability measure. The main result of the talk gives a multiplicative asymptotics for the deviation of ergodic sums for these automorphisms of Vershik.
The main element of the argument is the study of a special class of suspension flows over Vershik’s automorphisms. Applications include a multiplicative asymptotics of the ergodic sums for translation flows on flat surfaces.
A corollary yields limit theorems for such flows.
These results extend the earlier work of Anton Zorich on interval
exchange maps and are similar in spirit to that of Giovanni Forni on translation flows.