Shadowing, thick sets and the Ramsey property

Abstract

We provide a full characterization of relations between the shadowing property and the thick shadowing property. We prove that they are equivalent properties for non-wandering systems, the thick shadowing property is always a consequence of the shadowing property, and the thick shadowing property on the chain-recurrent set and the thick shadowing property are the same properties. We also provide a full characterization of the cases when for any family ${\mathcal{F}}$ with the Ramsey property an arbitrary sequence of points can be ${\it\varepsilon}$-traced over a set from ${\mathcal{F}}$.

Preservation of shadowing in discrete dynamical systems

“…Eventual shadowing was introduced in [23] in the authors' journey to characterise when the set of ω-limit sets of a system coincides with the set of closed internally chain transitive sets. As remarked upon in [23], the property of eventual shadowing is equivalent with the (N, F cf )-shadowing property of Oprocha [32].…”

“…Various other notions of shadowing have since been studied including, for example, ergodic, thick and Ramsey shadowing [8,9,14,17,32], limit shadowing [1,24,38], s-limit shadowing [1,24,27], orbital shadowing [23,38,36], and inverse shadowing [12,26].…”

On the shadowing and limit shadowing properties

“…To state another corollary of Theorem 1.1, we briefly recall the definition of the socalled thick shadowing property. Following the notation in [24], we define two families of subsets in N 0 = N ∪ {0} by…”

“…In [7], the thick shadowing property and several similar properties were proved to be equivalent to the shadowing property under the assumption of chain transitivity. The study was extended in [24], and a characterization of the thick shadowing property was given. According to [24,Theorem 4.5], a continuous map f : X → X has the thick shadowing property iff CR(f ) = Ω(f ) = R(f ) and f | Ω(f ) has the shadowing property, where R(f ) denotes the set of recurrent points for f , i.e., R(f ) = {x ∈ X : x ∈ ω(x)}.…”

Orbital shadowing, ω-limit sets and minimality

“…First used implicitly by Bowen [3], a system has shadowing, or the pseudo-orbit tracing property, if pseudo-orbits are shadowed by true orbits. Since then various other notions of shadowing have been studied, for example, ergodic, thick and Ramsey shadowing [4,5,8,10,19], limit shadowing [2,14,21], s-limit shadowing [2,14,17], orbital shadowing [12,21,20], and inverse shadowing [7,16].…”