On entropy, entropy-like quantities, and applications

This is a review on entropy in various fields of mathematics and science. Itsscope is to convey a unified vision of the classical entropies and some newer,related notions to a broad audience with an intermediate background in dynamicalsystems and ergodic theory. Due to the breadth and depth of the subject, wehave opted for a compact exposition whose contents are a compromise betweenconceptual import and instrumental relevance. The intended technical level andthe space limitation born furthermore upon the final selection of the topics,which cover the three items named in the title. Specifically, the first partis devoted to the avatars of entropy in the traditional contexts: manyparticle physics, information theory, and dynamical systems. Thischronological order helps present the materials in a didactic manner. Theaxiomatic approach will be also considered at this stage to show that, quite remarkably,the essence of entropy can be encapsulated in a few basic properties.Inspired by the classical entropies, further akin quantities have beenproposed in the course of time, mostly aimed at specific needs. A commondenominator of those addressed in the second part of this review is theirmajor impact on research. The final part shows that, along with its profoundrole in the theory, entropy has interesting practical applications beyondinformation theory and communications technology. For this sake we preferredexamples from applied mathematics, although there are certainly niceapplications in, say, physics, computer science and even social sciences. Thisreview concludes with a representative list of references.