Fluctuation bounds for ergodic averages of amenable groups

We study fluctuations of ergodic averages generated by actions of amenable groups. In the setting of an abstract ergodic theorem for locally compact second countable amenable groups acting on uniformly convex Banach spaces, we deduce a highly uniform bound on the number of fluctuations of the ergodic average for a class of Folner sequences satisfying an analogue of Lindenstrauss's temperedness condition. Equivalently, we deduce a uniform bound on the number of fluctuations over long distances for arbitrary Folner sequences. As a corollary, these results imply associated bounds for a continuous action of an amenable group on a sigma-finite Lp space with p is an element of(1,infinity).