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We study qualitative properties of the set of recurrent points of finitelygenerated free semigroups of measurable maps. In the case of a single generatorthe classical Poincare recurrence theorem shows that these properties areclosely related to the presence of an invariant measure. Curious, but otherwiseit turns out to be possible that almost all points are recurrent, while thereis an wandering set of positive (non-invariant) measure. For a generalsemigroup the assumption about the common invariant measure for all generatorslooks somewhat unnatural (despite being widely used). Instead we give abstractconditions (of conservativity type) for this problem and propose a weakerversion of the recurrent property. Technically, the problem is reduced to theanalysis of the recurrence of a specially constructed Markov process. Questionsof inheritance of the recurrence property from the semigroup generators to theentire semigroup and vice versa are studied in detail and we demonstrate thatthis inheritance might be rather unexpected.

Recurrence for measurable semigroup actions