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

The theory of zero-dimensional extensions is a useful tool in the study entropy theory. In present paper, we aim to which properties can be preserved by suitable extension. For dynamical system ( X , G ? ) prove that shadowing property implies it factor inverse limit subshifts with property. Moreover, if totally disconnected, extension conjugate. Besides, transitivity (or mixng) mixng, respecti...

We show that an invertible bilateral weighted shift is strongly structurally stable if and only it has the shadowing property. also exhibit a K{o}the sequence space supporting frequently hypercyclic shift, but no chaotic shifts.

We show that if a diffeomorphism has the periodic shadowing property on the chain recurrent set, then the closure of the periodic set is the chain recurrent set. Mathematics Subject Classification: 37C50

The orbital shadowing property (OSP) of discrete dynamical systems on smooth closed manifolds is considered. The nondensity of OSP with respect to the C1-topology is proved. The proof uses the method of skew products developed by Ilyashenko and Gorodetskĭı.