Recurrent sets and shadowing for finitely generated semigroup actions on metric spaces

We introduce various new type of recurrent sets for finitely generated semigroups on noncompact metric spaces that are conjugacy invariant, and obtain some basic properties of chain recurrent sets for semigroups via these new definitions. Moreover, we define the notion of weak shadowing property for finitely generated group actions on compact metric spaces, which is weaker than that of shadowing property, and prove the equivalence of the shadowing and weak shadowing properties for the finitely generated group actions on a generalized homogeneous space without isolated points.

Automated summary

This article introduces new types of recurrent sets on noncompact metric spaces for finitely generated semigroups that are conjugacy invariant, and explores the basic properties of chain recurrent sets for semigroups. Additionally, the notion of weak shadowing property for finitely generated group actions on compact metric spaces is defined, and the equivalence of shadowing and weak shadowing properties for group actions on a generalized homogeneous space without isolated points is proved.