A Non Homogeneous Ergodic Theorem

Abstract

Such series are of interest for several reasons. (1) In a paper with the same title as that of this one Izumi(2) has explicitly raised the problem of the convergence of (*) and offered a proof that, under certain rather restrictive conditions on f and T, (*) converges almost everywhere. (2) Very elegant necessary and sufficient conditions for the convergence of series similar to (*) are known in the theory of probability(')-an extension of those results would be of significance in the study of asymptotic properties of a more general class of transformations. (3) Since it is an elementary fact(4) that the convergence of the numerical series En=(1 /n)xn implies that limn_ (1 /n) {=1 Xi = 0, the almost everywhere convergence of (*) would be an analytic strengthening of Birkhoff's ergodic theorem. The principal result of this paper (stated precisely in ?3) is that in general (*) does not converge in the mean (of order two). 2. Izumi's theorem. Izumi assumes that the transformation T is uniformly mixing in the sense that