On origins of orbits and the shadow of chaos

Mitchell, Joel Stephen ORCID: 0000-0003-2659-8242 (2021). On origins of orbits and the shadow of chaos. University of Birmingham. Ph.D.

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Abstract

The work presented in this thesis concerns three areas within topological dynamics. The first area is that of chaos: specifically topological equicontinuity, transitivity, and sensitivity. We present two Auslander–Yorke dichotomy type theorems before constructing a chaotic system with an even continuity pair but no equicontinuity point. The second topic is an exploration of the preservation of various notions of shadowing under inverse limits, products, factor maps, and the induced maps for symmetric products and hyperspaces. The third area is that of \(\alpha\)-limit sets and \(\omega\)-limit sets in dynamical systems. These sets may be thought of as the origins and, respectively, destinations of orbit sequences. Many of our results on these sets relate to the aforementioned shadowing property and variations thereof. Included in these results is a characterisation of when the set of \(\alpha\)-limit sets, the set of \(\omega\)-limit sets, and the set of nonempty closed internally chain transitive sets (\(ICT_f\)) coincide. Moreover, we demonstrate that shadowing is sufficient to mean that every element of \(ICT_f\) can be approximated (to any prescribed accuracy) by both the \(\alpha\)-limit set and the \(\omega\)-limit set of the same full-trajectory.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

Supervisor(s):