TOPDS

Topological dynamics and chaos on compact metric spaces

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 Obiettivo del progetto (Objective)

'In the paper 'Measure of chaos and a spectral decomposition of dynamical systems of interval' (which extends Li and Yorke approach stated in their famous paper 'Period three implies chaos') Schweizer and Smital introduced the definition of distributional chaos. Scientific aim of this project is to study distributional chaos and its relations to other notions known from Topological Dynamics. Main problems we will cosider are the following: - how 'large' distributionally scrambled sets can be form topological, measure theoretic or dimension theory point of view? - what are sufficient conditions (topological mixing, specification property, topological exactness, shadowing) to ensure distributional scrambled sets being uncountable, perfect, invariant, etc. ? - what are condition not strong enough to imply distributional chaos in general case (e.g. it is known that positive topological entropy or weak mixing belongs to this class)? - are there any other spaces (graphs, dendrites, low-dimensional continua) which guarantee equivalent conditions from Schwaizer and Smital paper to hold (it is known that there is no equivalence in general, in particular in dimension two or zero)? Additionally, we will study shift spaces and their generalizations for a better understanding of the notion of 'complexity' in the theory of dynamical systems. The research undertaken in this project aims to extend knowledge about chaotic phenomena in dynamical systems. The main aim of the project is to extend knowledge and research experience of the researcher to the level that he is able to prepare his habilitation thesis. The researcher will present obtained results at international meetings. He will extend his scientific collaborations and start new independent lines of research in his career.'

Introduzione (Teaser)

An EU-funded research initiative supported researcher-directed study of topological dynamics and distributional chaos. Various project successes contributed to advances in this scientific area.

Descrizione progetto (Article)

The 'Topological dynamics and chaos on compact metric spaces' (TOPDS) project had as its scientific focus the study of distributional chaos and its relations to other notions of topological dynamics. The goal was to extend knowledge about chaotic phenomena in dynamical systems, and at the same time to expand the knowledge and research experience of the researcher.

Over the course of the project, the researcher achieved a number of successful results, which were to be presented at international meetings. He was also tasked with extending scientific collaborations and starting on new independent lines of research.

The technique used to prove that it is possible to transfer a distributionally scrambled set from factor to extension was also used to investigate the dynamics of non-autonomous differential equations. Other TOPDS successes include the development of a formal method of measuring complexity of these equations which, in this context, provided a strict edge between chaotic and non-chaotic dynamics.

Activities also resulted in delivery of a method for constructing continuous maps and obtaining elementary proofs relevant to dense periodicity for maps on topological graphs and on specific spaces.