Pointwise Pseudo-Orbit Tracing Property and Its Application

Abstract:

      In this paper, the pointwise pseudo-orbit tracing property is defined on a compact metric space, and it is a generalization of pseudo-orbit tracing property. As applications,we prove the following results: (i) If f has pointwise pseudo-orbit tracing property, for any k ∈ Z₊, and f^k is chain transitive, then for any k ∈ Z₊, f^k has open set transitive; (ii)If f has pointwise pseudo-orbit tracing property, and for any n ∈ Z₊,fⁿ is chain transitive,then f has sensitive dependence on initial conditions; (iii) If f is open set mixing and haspointwise pseudo-orbit tracing property, then f has property P; (iv) Let f: (X, d) → (X, d)be a homeomophism, then f is pointwise pseudo-orbit tracing property if and only if the shift map σ_f is pointwise pseudo-orbit tracing property.