Recurrence for measurable semigroup actions

Abstract

We study qualitative properties of the set of recurrent points of finitely generated free semigroups of measurable maps. In the case of a single generator the classical Poincare recurrence theorem shows that these properties are closely related to the presence of an invariant measure. Curious, but otherwise it turns out to be possible that almost all points are recurrent, while there is an wandering set of positive (non-invariant) measure. For a general semigroup the assumption about the common invariant measure for all generators looks somewhat unnatural (despite being widely used). Instead we give abstract conditions (of conservativity type) for this problem and propose a weaker version of the recurrent property. Technically, the problem is reduced to the analysis of the recurrence of a specially constructed Markov process. Questions of inheritance of the recurrence property from the semigroup generators to the entire semigroup and vice versa are studied in detail and we demonstrate that this inheritance might be rather unexpected.1. Introduction. Recall that a (discrete time) semigroup of maps T is a closed under composition collection of maps acting on the same topological space X. One says that T is finitely generated if there is a finite number of elements T := {T 1 , T 2 , . . . , T d } ⊂ T whose compositions generate T . Given a point x ∈ X one can define its trajectory under the action of the semigroup T in two very different ways. First of them, to which we will refer as a global trajectory, is the union of all images of the point x under the action of the maps from T , i.e. ∪ T ∈T T x. The main disadvantage of this definitions is that the points of the global trajectory are not ordered. To avoid this difficulty we consider another approach, to which we will refer as a realization (as in random processes), and which can be described as an infinite sequence of compositions of the generators of the semigroup evaluated at the pointThe union of all realizations is exactly the global trajectory of the point under the action of the semigroup.The idea of recurrence is very old and simple. Roughly speaking it means that a trajectory (or a realization of a random process) returns eventually arbitrary close to its initial point. From the point of view of dynamical systems theory, recurrence is a distant generalization of the notion of periodicity. In terms of the global trajectories this means that there is a sequence {τ j1 , . . . , τ jn , . . . } of elements τ j ∈ T such that