## Product Type Dynamical Systems: variational principle

(English Below)

Este é um resumo do seminário que vou apresentar dia 26 de janeiro de 2012 no Imperial College London.

This is the abstract of a seminar I am about to present on 26 January of 2012 at the Imperial College London DynamIC Seminars. The slides will be upload to this page as soon as they are available.

Comments are very welcome, and can be made at the end of this page.

### Abstract

Inspired by the Kolmogorov-Sinai entropy (KS-entropy) for a measure-preserving dynamical system, Adler, Konheim and McAndrew developed a purely topological concept of entropy (AKM-entropy) for topological dynamical systems over compact phase spaces. The AKM-entropy relates to the KS-entropy trough the so called Variational Principle: $h(T) = \sup_\mu h_\mu(T),$ where $h(T)$ is the AKM-entropy and $h_\mu(T)$ is the KS-entropy for the dynamical system $T$. The supremum is taken over all possible Borel measures.

Since then, many attempts to generalise this to non-compact spaces have been made. But not always the Variational Principle holds for this new concept. Bowen did it for metrizable systems. For a definition that uses some heavy machinery under the umbrella of topological pressure, Pesin and Pitskel’ have proved that the Variational Principle holds under some (hard to verify) hypothesis. In this Seminar, I will present a new concept of entropy which is surprisingly close to the AKM-entropy, and for which the Variational Princilpe still holds for a wide range of spaces. We have called those, Product Type Dynamical Systems.