Topological dynamics

Lead Research Organisation: University of Birmingham
Department Name: School of Mathematics

Abstract

This project looks at discrete dynamical systems from a topological view point. We start by reviewing key literature. After that the project will focus on (i) dynamical properties from a non-metric topological view point, (ii) shadowing and stability in relation to omega limit sets and (iii) the structure of alpha limit sets.

Publications

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Good C (2020) Preservation of shadowing in discrete dynamical systems in Journal of Mathematical Analysis and Applications

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Good C (2020) On inverse shadowing in Dynamical Systems

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Mitchell J (2019) Orbital shadowing, ?-limit sets and minimality in Topology and its Applications

Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509590/1 01/10/2016 30/09/2021
1809312 Studentship EP/N509590/1 01/10/2016 30/03/2020 Joel Stephen Mitchell
 
Description Several important results in dynamics have been established as a result of the work funded through this award.

In topological dynamics, one is often concerned with where points come from (the alpha limit sets) and where they end up (the omega limit sets). We establish that for maps with a property called shadowing, any internally chain transitive set can be approximated by the alpha and omega limit sets of the same full-trajectory.

The Auslander-Yorke dichotomy is a well-known result in dynamics concerning minimal maps on compact metric spaces. We generalise this result in a couple of ways to more general settings. We also introduce a new concept: eventual sensitivity. Sensitivity, a well-established property in dynamics and chaos theory, means that, no matter where one starts, there are two points arbitrarily close to each other and to that starting location which will move far apart as time progresses; a universal `far'. Clearly this is extremely relevant in an applied setting; rounding errors mean a computer will not, generally, track true orbits. But what if every point moves arbitrarily close to another point that it will then move away from? What if a computer starts with a true orbit and tracks it accurately, but then the point moves close to another point which will end up going in completely the other direction? - these two points may be so close together that the computer cannot differentiate between them; it may start tracking the wrong orbit and give an extremely inaccurate prediction of the future. It is this idea that motivated the notion of eventual sensitivity - a concept which has now attracted the attention of other academics.

Elsewhere we study the pseudo-orbit tracing (or shadowing) properties and consider how and if they are preserved in induced dynamical systems. These results should prove a useful resource for those working with dynamical systems since shadowing has been shown to have both theoretical and numerical importance.

Finally we show that pseudo-orbits (i.e. orbits but with `rounding errors') trap true orbits in a neighbourhood of prescribed accuracy after a uniform time period.
Exploitation Route As with many results in pure mathematics, the future outcomes and uses of many results here are difficult to ascertain. The results here will certainly be beneficial to the topological dynamics community, and, it is hoped, used by those working in applied dynamics.
Sectors Education,Other