Recurrence for measurable semigroup actions

We study qualitative properties of the set recurrent points finitely generated free semigroups measurable maps. In case a single generator classical Poincare recurrence theorem shows that these are closely related to presence an invariant measure. Curious, but otherwise it turns out be possible almost all recurrent, while there is wandering positive (non-invariant) For general semigroup assumption about common measure for generators looks somewhat unnatural (despite being widely used). Instead we give abstract conditions (of conservativity type) this problem and propose weaker version property. Technically, reduced analysis specially constructed Markov process. Questions inheritance property from entire vice versa studied in detail demonstrate might rather unexpected.