ALMOST SURE ASYMPTOTIC BEHAVIOUR OF BIRKHOFF SUMS FOR INFINITE MEASURE-PRESERVING DYNAMICAL SYSTEMS

We consider a conservative ergodic measure-preserving transformation T of a sigma-finite measure space (X; B; mu) with mu(X) = infinity. Given an observable f : X -> R, we study the almost sure asymptotic behaviour of the Birkhoff sums S(N)f(x) := Sigma(N)(j=1)(f omicron Tj-1)(x). In infinite ergodic theory it is well known that the asymptotic behaviour of S(N)f (x) strongly depends on the point x is an element of X, and if f is an element of L-1 (X; mu), then there exists no real valued sequence (b(N)) such that lim N!1 SNf (x)=b(N) = 1 almost surely. In this paper we show that for dynamical systems with strong mixing assumptions for the induced map on a finite measure set, there exists a sequence (alpha(N)) and m : X x N -> N such that for f is an element of L-1 (X, mu) we have lim(N ->infinity)1 S(N+m(x,N))f (x)/alpha(N) = 1 for mu-a.e. x is an element of X. Instead in the case f is not an element of L-1 (X, mu) we give conditions on the induced observable such that there exists a sequence (G(N)) depending on f, for which lim(N ->infinity) S(N)f (x)/G(N) = 1 holds for mu-a.e. x is an element of X.