Preservation of shadowing in discrete dynamical systems

Abstract

We look at the preservation of various notions of shadowing in discrete dynamical systems under inverse limits, products, factor maps and the induced maps for symmetric products and hyperspaces. The shadowing properties we consider are the following: shadowing, h-shadowing, eventual shadowing, orbital shadowing, strong orbital shadowing, the first and second weak shadowing properties, limit shadowing, s-limit shadowing, orbital limit shadowing and inverse shadowing.

Shadowing, internal chain transitivity and α-limit sets

“…In this case we say z shadows or ε-shadows the sequence x i ∞ i=0 . Shadowing has both numerical and theoretical importance and has been studied extensively in a variety of settings; in the context of Axiom A diffeomorphisms [9], in numerical analysis [14,15,37], as an important factor in stability theory [40,43,47], in understanding the structure of ω-limit sets and Julia sets [5,6,7,10,33], and as a property in and of itself [16,24,26,31,35,38,40,44]. A variety of variants of shadowing have also been studied including, for example, ergodic, thick and Ramsey shadowing [11,12,19,21,36], limit, or asymptotic, shadowing [4,27,41], s-limit shadowing [4,27,31], orbital shadowing [23,34,39,41], and inverse shadowing [15,25,30].…”

On inverse shadowing

“…Such classes have been studied in a variety of different settings for example [18,19,20,22,29,33]. Of particular interest has been its relationship to structural stability.…”

Invariants Of Uniform Conjugacy On Uniform Dynamical System

“…Given , a -psendo-trajectory * + is called -traced by , if ( ( ) ) , ( ). Here the symbols and are takenas , if is bijective and as , -, [3]. If is not bijective, we say that has the shadowing property or (pseudo-trajectory tracing property) for every there exists such that every -pseudo-trajectory of can be traced by some point of .…”