Poincaré Return Times as Universal Sequences

Let (X, 36, m) be a probability space and let x be any invertible measure-preserving transformation of X. Given A, Bs 08, the Poincare return time sequence is the sequence of whole numbers n(A,B) = (neZ: m(A n zB) > 0). There is also a related point return time sequence, given for xeX and A €08 by n(x,A) = (neZ: xxsA). A Poincare return time sequence is the democratic version of the point return time sequences. Both types of return time sequences have been studied extensively in ergodic theory. In particular, recently the point return time sequence has been shown by Bourgain [3] to be a universally good sequence for a.e. x. See also Bourgain [4], where a proof of this theorem by Furstenberg, Katznelson and Ornstein is given. In Bellow and Losert [1], further background material on return time sequences and universal sequences is discussed. In this article, the issue considered is whether the Poincare return time sequences are universally good. Given a sequence n = (nk: k= 1,2,3,...), the sequence is said to be Lqpointwise universally good if for all dynamical systems (Y, T,p,a), and all functions feLq(Y), the averages (l/N) Yjk-\A) converge a.e. [p]. The sequence is said to be Lq norm universally good if for all dynamical systems (Y, T,p,a), and all functions/eLQ(Y), the averages (l/N) ££L1./(a *iy) converge in Lg-norm. It is clear that Lq pointwise universally good sequences are norm universally good for any Lr, 1 < r < oo. Also, it is well-known that L2 norm universally good sequences are exactly the ones such that for all x, linijv^ (1/AO YJk-\ P(**) exists. It is not hard to see that if 1 ^ q < oo, any Lq norm universally good sequence is Lr norm universally good for any r, 1 ^ r < oo. Thus, norm universality is just a trigonometric property of the sequence. The issue of whether or not a Poincare return time sequence is universally good is worthwhile considering only for compact dynamical systems. Indeed, if x is strongly mixing, then n(A,B) contains all n 5* N for some N depending on A and B. Also, if x is just weakly mixing, then n(A, B) has density one. Consequently, in these cases, it is clear that the Poincare return time sequences are universally good in all senses. However, for a compact dynamical system, like an irrational rotation of the circle group, the universality of n(A,B) is a very different issue. See Bergelson and Rosenblatt [2] for a description of compact dynamical systems. Let Gbea compact abelian group and let g be a generator of G. With m being the Haar measure on G, and x(x) = gx, we have the typical compact dynamical system. The first goal here is to show that for a.e. heG, the sequence n(A, hB) is a universally good sequence in all senses. To facilitate proving this result, we need some definitions.