Ergodic theorems for amenable actions

Abstract

Classical ergodic theory deals with the action of N on an arbitrary ¯nite measure space (X; ¹). It is long recognized that many of the results in ergodic theory can be generalized to more general acting groups. The appropriate class of groups has turned out to be the class of amenable groups. It has taken remarkably long to extend the pointwise ergodic theorem to the action of any amenable group. Although this result was extended to many particular subclasses of amenable groups, the full result was established only in [12]. The result relies on a weak maximal inequality for the action of G. The extension of the Cesµaro averages to amenable groups can be viewed as a function that is similar to the maximal function, introduced by Hardy and Littlewood. This induces an operator A¤, that assigns this maximal function to f 2 Lp(X). We will show that this operator is bounded, whenever 1 < p < 1. To do so, we will ¯rst prove the statement for the action of G on itself. Thereafter, we derive it for the action of G on every measure space, using a technique known as transference.