On Notions of Entropy in Physics and Mathematics | Chapter 11 | Research Trends and Challenges in Physical Science Vol. 7

 The pair (X,f) denotes discrete topological dynamical systems, where X is a topological space and f:XXX is a continuous map. Throughout the years, a lengthy number of results have emerged to understand and make sense of what the complexity of the systems is. Topological entropy is a helpful technique, and one of the most widely used. In most applications, the phase space X is a small metric space. Other X and f conditions have also been considered. For instance, X could be non-compact or f could 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 of theoretical interest in Topological Dynamics, as well as in applied fields such as Electronics and Control Theory.

Entropies are the topic of this paper. We begin with the initial conceptions of entropy in thermodynamics and follow their evolution until the appearance in the twentieth century of the notions of Shannon and Kolmogorov-Sinai entropies, and the following notion of topological entropy influenced by them. In the mathematical context, such conceptions have evolved, with extended versions appearing to cover modern concerns.

Author(S) Details

Francisco Balibrea
University of Murcia, Spain.

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