Coupling of the GB set property for ergodic averages

Abstract

Let (Y,ℬ,μ,T) be an ergodic dynamical system. LetA be an nonempty subset ofL ²(μ) such that$I(A) = \int_0^{diam(A)} {\sqrt {\log N(A,u)} du< \infty }$, whereA=sup{||sȒt||_2μ ,s, t∈A} andN(A, u) is the smallest number ofL ²(μ)-open balls of radiusu, centered inA, enough to coverA. Let$C(A) = \left\{ {\tfrac{1}{n}\Sigma _{i - 0}^{n - 1} f \circ T^i ,n \geqslant 1,f \in A} \right\}$. We prove as a consequence of a more general result, thatC(A) is aGB subset ofL ²(μ).