APOTP And Recurrent Property

In this paper,we introduce a new concept-the asymptotic pseudo orbit tracing property(abbrev.APOTP),and mainly discuss the fundamental properties of the APOTP and the relationship between APOTP with recurrent property.In the first section,we introduce some notions and known results about topo-logical dynamic system.In the second section,we introduce the concept of APOTP and show that the APOTP is invariant under both the iterates of map and the topological conjugacy.In the third section,we study the relationship between the APOTP for continuous map / on a compact metric space X and that for the lift map / of / on a covering space X and show that (X, f) has the APOTP if and only if (X, /) has the APOTP.In the fourth section,we study the connection between the APOTP for continuous map on a compact metric space and that for the shift map on the inverse limit space.In the fifth section,we study transitive properties and recurrent properties of a system having the asymptotic pseudo orbit tracing property. In particular, we obtain a condition for which a homeomorphism is a non-wanderinghomeomorphism.