On the Shadowing Property and Shadowable Point of Set-valued Dynamical Systems
- PDF / 228,670 Bytes
- 11 Pages / 510.236 x 737.008 pts Page_size
- 67 Downloads / 213 Views
Acta Mathematica Sinica, English Series Springer-Verlag GmbH Germany & The Editorial Office of AMS 2020
On the Shadowing Property and Shadowable Point of Set-valued Dynamical Systems Xiao Fang LUO
Xiao Xiao NIE
Jian Dong YIN1)
Department of Mathematics, Nanchang University, Nanchang 330031, P. R. China E-mail : [email protected] [email protected] [email protected] Abstract In this article, the authors introduce the concept of shadowable points for set-valued dynamical systems, the pointwise version of the shadowing property, and prove that a set-valued dynamical system has the shadowing property iff every point in the phase space is shadowable; every chain transitive set-valued dynamical system has either the shadowing property or no shadowable points; and for a set-valued dynamical system there exists a shadowable point iff there exists a minimal shadowable point. In the end, it is proved that a set-valued dynamical system with the shadowing property is totally transitive iff it is mixing and iff it has the specification property. Keywords
Shadowing property, shadowable point, set-valued dynamical system, specification
MR(2010) Subject Classification
1
54H20, 37D45
Introduction
For convenience, we denote by N the set of positive integers, N0 the set of non-negative integers and Z the set of integers. For a discrete dynamical system (X, f ), where X is a compact metric space with a metric d and f : X → X is a continuous map, the shadowing property was introduced by Bowen [3]. Fix δ > 0. An infinite sequence ξ = {ξn }n∈N0 of X is called a δ-pseudo-orbit of f if d(f (ξn ), ξn+1 ) ≤ δ for all n ∈ N0 . (X, f ) or f is said to have the shadowing property if for any > 0, there exists δ > 0 such that every δ-pseudo-orbit {xi }i∈N0 of f can be -traced by some point in X, i.e., there exists some y ∈ X such that d(f i (y), xi ) < for each i ∈ N0 . In 2016, Morales [12] introduced the notion of shadowable points for homeomorphisms of compact metric spaces which is referred to as the pointwise inversion of the shadowing property of discrete dynamical systems. More precisely, let g : X → X be a homeomorphism of a compact metric space X and δ > 0. An infinite sequence ξ = {ξn }n∈Z of X is called a δ-pseudo-orbit of g if d(g(ξn ), ξn+1 ) ≤ δ for all n ∈ Z. A point x ∈ X is called a shadowable point of g if for every > 0 there exists δ := δ(, x) > 0 such that every δ-pseudo-orbit ξ = {ξn }n∈Z of g through {x} (i.e., ξ0 = x) can be -shadowed by some point in X, i.e., there exists y ∈ X such that d(f i (y), xi ) < for each i ∈ Z. Denote by Sh(g) the set of all shadowable points of g. Morales [12] proved that Sh(g) is g-invariant, i.e., g(Sh(g)) = Sh(g), and Sh(g) may be Received August 16, 2019, accepted July 1, 2020 Supported by the National Natural Science Foundation of China (Grant Nos. 11661054, 11261039) 1) Corresponding author
Shadowable Points of Set-valued Dynamical Systems
1385
non-empty and non-compact. See [9, 10] for more detailed results of shadowable points and see [2] for shadowable points of flows. Recent
Data Loading...
