ON SYSTEMS WITH FINITE ERGODIC DEGREE

Abstract

In this paper we study the ergodic theory of a class of symbolic dynamical systems where is the left shift transformation on and µ is a σ-finite T-invariant measure having the property that one can find a real number d so that but for all where t is the first passage time function in the reference state 1. In particular, we shall consider invariant measures m arising from a potential V which is uniformly continuous but not of summable variation. If then m can be normalized to give the unique non-atomic equilibrium probability measure of V for which we compute the (asymptotically) exact mixing rate, of order We also establish the weak-Bernoulli property and a polynomial cluster property (decay of correlations) for observables of polynomial variation. If instead then m is an infinite measure with scaling rate of order Moreover, the analytic properties of the weighted dynamical zeta function and those of the Fourier transform of correlation functions are shown to be related to one another via the spectral properties of an operator-valued power series which naturally arises from a standard inducing procedure. A detailed control of the singular behaviour of these functions in the vicinity of their non-polar singularity at z = 1 is achieved through an approximation scheme which uses generating functions of a suitable renewal process. In the perspective of differentiable dynamics, these are statements about the unique absolutely continuous invariant measure of a class of piecewise smooth interval maps with an indifferent fixed point.