Entropy in asymmetric topological structures

Abstract

The notion of entropy rst appeared in the context of thermodynamics in the rst half of the XIX century. The topological entropy and the uniform entropy have been largely studied in the last 60 years. The main aim of this thesis is to investigate the notion of entropy in asymmetric spaces. More speci cally, we are interested in extending the notion of entropy on metric and uniform spaces to quasi-metric and quasi-uniform spaces. In this thesis we managed to generalize most of the results about uniform entropy on metric(uniform) spaces to quasi-metric(uniform) spaces. Indeed a new notion of entropy for a uniformly continuous self-map of quasi-metric(uniform) space has been presented, which we call quasi-uniform entropy in this thesis. We managed to show that the quasi-uniform entropy is less or equal to the uniform entropy of considered as a uniformly continuous self-map of the metric(or uniform) space (X; qs) (or (X; Us)), where qs( or Us) is the symmetrised metric(or uniformity) of the quasi-metric(or uniformity) q( or U). We prove that for a join-compact quasi-metric(uniform) space the quasi-uniform entropy of a uniformly continuous self-map coincides with the quasi-uniform entropy of its extension to the bicompletion. Finally, we compared our notion of entropy namely, quasi-uniform entropy, to some well-known notions of topological entropy.