ZERO-ONE LAW OF HAUSDORFF DIMENSIONS OF THE RECURRENT SETS

Abstract

Let (Sigma, sigma) be the one-sided shift space with m symbols and R-n(x) be the first return time of x is an element of Sigma to the n-th cylinder containing x. Denote E-alpha, beta(phi) - {x is an element of Sigma : lim(n ->infinity) inf log R-n(x)/phi(n) - alpha, lim(n ->infinity) sup log R-n(x)/phi(n) - beta}, where phi : N -> R+ is a monotonically increasing function and 0 <= alpha <= beta <=+infinity. We show that the Hausdorff dimension of the set E-alpha, beta(phi) admits a dichotomy: it is either zero or one depending on phi, alpha and beta.