Suspension Flows over Vershik’s Automorphisms

The aim of this paper is to give a multiplicative asymptotics for the deviation of ergodic averages for certain classes of suspension flows over Vershik’s automorphisms [11, 13]. A corollary of the main result yields limit theorems for these flows. Informally, Vershik’s automorphisms (also sometimes called “adic transformations”) are dynamical systems whose orbits are leaves of the asymptotic foliation of a Markov chain. Two cases are considered in this paper: that of a time-homogeneous Markov chain (when the corresponding automorphisms are called “periodic”) and that of a Markov chain whose adjacency matrices are given by a stationary law. A particular instance of the second situation is given by generic interval exchange transformations, whereas, if an interval exchange transformation is a periodic point of the Rauzy-Veech-Zorich induction map, then we find ourselves in the first situation. By the Vershik-Livshits Theorem [13], another example of periodic Vershik’s automorphisms is furnished by subshifts corresponding to primitive substitutions. We further consider bi-infinite Markov chains and introduce flows whose orbits are leaves of the asymptotic foliation. These flows are suspension flows over Vershik’s automorphisms, with a roof function assuming finitely many values. A particular case of these flows is given by translation flows on flat surfaces. The case of a time-homogeneous Markov chain includes flows along stable foliations of pseudo-Anosov automorphisms. To study the asymptotics of ergodic averages, we introduce the space of additive continuous holonomy-invariant functionals on the orbits of our flows. Informally, these functionals are dual objects to invariant distributions of G. Forni [3, 4]. The space of these functionals is finite-dimensional, and they are given by an explicit construction. The main result of the paper (Theorem 1 in the periodic case; Theorem 2 in the general case) states that the time integrals of Lipschitz functions can be approximated by these functionals up to an error which grows slower than any power of the time. It follows (Corollary 3 in the periodic case; Corollary 5 in the general case) that time integrals of Lipschitz functions, taken at an