Recurrence in mean for group actions

Abstract

In this paper, the mean values of the recurrence are computed for general group actions. Let [Formula: see text] be a metric space with a finite measure [Formula: see text] and [Formula: see text] be a countable group acting on [Formula: see text]. Let [Formula: see text] be a sequence of subsets of [Formula: see text] with [Formula: see text] and put [Formula: see text]. If the Hausdorff measure [Formula: see text] is finite on [Formula: see text] and [Formula: see text] is [Formula: see text]-invariant. We assume that [Formula: see text] and [Formula: see text] are concordant. Then the function [Formula: see text] is [Formula: see text]-integrable and for any [Formula: see text]-measurable set [Formula: see text] we have [Formula: see text] If moreover, [Formula: see text] then [Formula: see text] without the concordance condition for the measure [Formula: see text] and [Formula: see text]