When is the beginning the end? On full trajectories, limit sets and internal chain transitivity

2020/04/05 by J. B. Mitchell, Mitchell, Joel
Mathematics · #37B99 #37C50 #54H20 #Dynamical Systems (math.DS) #FOS: Mathematics #Fixed Point Theorems Analysis #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2004.02878

openalex publication_date 2020/04/05 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

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. In this paper we characterise, by introducing novel variants of shadowing, maps for which every element of ICT_f is equal to (resp. may be approximated by) the α-limit set and the ω-limit set of the same full trajectory. We construct examples highlighting the difference between these properties.

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