POINTWISE CONVERGENCE OF ERGODIC AVERAGES IN ORLICZ SPACES

摘要

We construct a sequence a_n such that for any aperiodic measure-preserving system (X,Σ,m,T) the ergodic averages A_N f(x) = 1/N Σ_(n-1)~N f(T~(a,n x)) converge a.e. for all / in L log log(L) but fail to have a finite limit for an f ∈ L~1. In fact, we show that for each Orlicz space properly contained in L~1 there is a sequence along which the ergodic averages converge for functions in the Orlicz space, but diverge for all f ∈ L~1. Our method, introduced by A. Bellow and extended by K. Reinhold and M. Wierdl, is perturbation.