arXiv subjects

Robert Leek

Publications and source records attributed to Robert Leek.

3 recordsLinked to original sources

arXiv2019

Equicontinuity, Transitivity and Sensitivity: the Auslander-Yorke Dichotomy Revisited

We discuss topological equicontinuity and even continuity in dynamical systems. In doing so we provide a classification of topologically transitive dynamical systems in terms of equicontinuity pairs, give a generalisation of the Auslander-Yorke Dichotomy for minimal systems and show there exists a transitive system with an even continuity pair but no equicontinuity point. We define what it means for a system to be eventually sensitive; we give a dichotomy for transitive dynamical systems in relation to eventual sensitivity. Along the way we define a property called splitting and discuss its relation to some existing notions of chaos.

math.DS↗

arXiv2015

Convergence properties and compactifications

In this paper, we will use investigate the existence of compactifications with particular convergence properties - pseudoradial, radial, sequential and Fréchet-Urysohn - through the use of spoke systems.

math.GN↗

arXiv2014

An internal characterisation of radiality

In this paper, we will investigate how radiality occurs in topological spaces by considering neighbourhood bases generated by nests. We will define a new subclass of radial spaces, called the independently-based spaces, that contains LOTS, GO-spaces and spaces with well-ordered neighbourhood bases. We show that first-countable spaces are precisely the independently-based, strongly Fréchet spaces and we give an example of a Fréchet-Urysohn space that is neither independently-based nor strongly Fréchet.

math.GN↗