Shadowing, internal chain transitivity and α-limit sets

2019/09/06 by Good, Chris, Meddaugh, Jonathan, Mitchell, Joel · 1 citation
#37B99 #54H20 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1909.02881

Abstract

Let f \colon X → X be a continuous map on a compact metric space X and let α_f, ω_f and ICT_f denote the set of α-limit sets, ω-limit sets and nonempty closed internally chain transitive sets respectively. We show that if the map f has shadowing then every element of ICT_f can be approximated (to any prescribed accuracy) by both the α-limit set and the ω-limit set of a full-trajectory. Furthermore, if f is additionally c-expansive then every element of ICT_f is equal to both the α-limit set and the ω-limit set of a full-trajectory. In particular this means that shadowing guarantees that α_f=ω_f=ICT(f) (where the closures are taken with respect to the Hausdorff topology on the space of compact sets), whilst the addition of c-expansivity entails α_f=ω_f=ICT(f). We progress by introducing novel variants of shadowing which we use to characterise both maps for which α_f=ICT(f) and maps for which α_f=ICT(f).

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