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Chaos theory from the mathematical point of view

Dominik Kwietniak;  Piotr Oprocha — 2008

Mathematica Applicanda

This work is intended as an attempt to survey existingde finitions of chaos for discrete dynamical systems. Discussion is restricted to the settingof topological dynamics, while the measure-theoretic (ergodic theory) and smooth (differentiable dynamical systems) aspects are omitted as exceedingt he scope of this paper. Chaos theory is understood here as a part of topological dynamics, so aforementioned definitions of chaos are just examples of particular dynamical system properties, and are considered...

On almost specification and average shadowing properties

Marcin Kulczycki;  Dominik Kwietniak;  Piotr Oprocha — 2014

Fundamenta Mathematicae

We study relations between the almost specification property, the asymptotic average shadowing property and the average shadowing property for dynamical systems on compact metric spaces. We show implications between these properties and relate them to other important notions such as shadowing, transitivity, invariant measures, etc. We provide examples showing that compactness is a necessary condition for these implications to hold. As a consequence, we also obtain a proof that limit shadowing in...

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