Orbital Shadowing Property

Abstract

Abstract. Let M be a generalized homogeneous compact space, and let Z(M ) denotes the space of homeomorphisms of M with the C 0 topology. In this paper, we show that if the interior of the set of weak stable homeomorphisms on M is not empty then for any open subset W of Z(M ) containing only weak stable homeomorphisms the orbital shadowing property is generic in W .

Weak Strictly Persistence Homeomorphisms and Weak Inverse Shadowing Property and Genericity

“…Let > 0 be arbitrary and U = {U i : i = 1, · · · , k} be a finite covering of M by open sets with diameter less than or equal 2 . As in the proof of Proposition 2 in [3], for each x ∈ M there exits δ x > 0 and N x ∈ N such that for each y ∈ N δx (x) and…”

“…We say that f is weak stable if every point of M is a weak stable point for f . The authors in [3] showed that for a homeomorphism f on compact metric space M , the set of weak stable points is residual in M . Moreover if f is minimal, then f is a weak stable homepmorphism.…”

Weak stability of non-autonomous discrete dynamical systems

“…Along this line, the study of shadowing property in autonomous dynamical systems attracts lots of attention [7, 8, 10, 11, 12, 13, 14, and the references therein]. In [9], a concept of weak stability has been introduced, and it is shown that orbital shadowing property is generic in the set of weak stable homeomorphisms. Motivated by this idea, we discuss weak stability in nonautonomous dynamical systems.…”