  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.
