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.
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: where is the AKM-entropy and is the KS-entropy for the dynamical system . 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.
- Preprint: André Caldas & Mauro Patrão. Entropy and Its Variational Principle for Product Type Dynamical Systems.
- Slides: Presentation PDF.