On genericity of shadowing in one dimension

Abstract

We show that shadowing is a generic property among continuous maps and surjections on a large class of locally connected one-dimensional dynamical systems.

Shadowing, recurrence, and rigidity in dynamical systems

“…As mentioned in the introduction, there is a significant body of work devoted to determining the prevalence of shadowing in C(X) for various categories of topological spaces X. In particular, shadowing has been shown to be a generic property of dynamical systems on manifolds ( [23,17,20,15,11]), dendrites ( [2]), and more exotic locally connected continua ( [12]). In addition, shadowing has been shown to be generic in the space of surjections (S(X)) on spaces in those classes.…”

“…In particular, in [12] we asked whether there were, in fact any continua X such that shadowing is not generic in C(X).…”

“…The shadowing property has wide-reaching applications in the analysis of dynamical systems, especially in analyzing stability [19,21,22] and in characterizing ω-limit sets [1,14]. Despite being a relatively strong property, shadowing is a generic property in the space of dynamical systems on many spaces [2,11,12,20].…”

Shadowing as a Structural Property of the Space of Dynamical Systems

“…Since then, it has been observed that shadowing plays an important role in stability theory [21,23,25] and in characterizing ω-limit sets [1,2,14]. Shadowing has also been shown to be a relatively common property in the space of dynamical systems on certain classes of spaces [5,11,13,15,16,22,26].…”

Recurrence, rigidity, and shadowing in dynamical systems

“…The shadowing property has wide-reaching applications in the analysis of dynamical systems, especially in analyzing stability [21,23,24] and in characterizing ω-limit sets [3,16]. Despite being a relatively strong property, shadowing is a generic property in the space of dynamical systems on many spaces [4,12,14,22].…”