On recurrence

Abstract

Let T be a non-singular ergodic automorphism of a Lebesgue space (X, L, μ) and let f: X→ℝ be a measurable function. We define the notion of recurrence of such a function f and introduce the recurrence set R(f)={α∈ℝ:f-α is recurrent}. If {Mathematical expression}, then R(ρ)={0}, but in general recurrence sets can be very complicated. We prove various conditions for a number α∈ℝ to lie in R(f) and, more generally, for R(f) to be non-empty. The results in this paper have applications to the theory of random walks with stationary increments. © 1984 Springer-Verlag.