When is the beginning the end?

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• Joel Mitchell (University of Birmingham)
• Thursday 18 March 2021, 15:00-16:00
• Zoom.

If you have a question about this talk, please contact David Craven.

Let f : 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. α‑ and ω‑limit sets may be viewed as the beginnings and ends of orbit sequences. 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. In particular this means that the presence of shadowing guarantees that ᾱ_f = ῶ_f = ICT _f (where the closures are taken with respect to the Hausdorff topology on the space of compact sets). We progress by introducing a property which characterises when all beginnings are ends of all beginnings, and all ends, beginnings of all ends.

This talk is partly based on a joint work with C. Good and J. Meddaugh.

This talk is part of the Topology and Dynamics seminar series.

This talk is included in these lists:

• School of Mathematics events
• Topology and Dynamics seminar
• Zoom

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