Minimal rates of entropy convergence for completely ergodic systems

If (X,T) is a completely ergodic system, then there exists a positive monotone increasing sequence {an}(n=1)(infinity) with lim(n-->infinity) a(n) = infinity and a positive concave function g defined on [1, infinity) for which g(x)/x(2) is not integrable such that [GRAPHICS] for all nontrivial partitions alpha of X into two sets.