On Notions of Entropy in Physics and Mathematics

Abstract

Discrete topological dynamical systems are given by the pair (X,f) where X is a topological space and f:X\(\rightarrow \)X a continuous maps. During years, a long list of results have appeared to understand and give sense to what is the complexity of the systems. Among others a useful tool and one of the most popular is that of topological entropy. The phase space X in most applications is a compact metric space. Even other conditions on X and f have been considered. For example X can be non-compact or f can be discontinuous (only in a finite number of points and with bounded jumps on the values of f or even unbounded jumps). Such systems are interesting from theoretical point of view in Topological Dynamics and by applications in applied sciences such as Electronics and Control Theory. In this paper we are dealing with entropies. We start with the original ideas of entropy in Thermodinamics and their evolution until the appearing in the twenty century of the notions of Shannon and Kolmogorov-Sinai entropies and the subsequent notion inspired on them of topological entropy. In the mathematical setting such notions have evolved apppearing extended versions to cover recent problems.