Prof. Yakov B. Pesin (The Pennsylvania State University), Equilibrium measures in hyperbolic dynamics via geometric measure theory

In the classical settings of Anosov diffeomorphisms or more general locally maximal hyperbolic sets I describe a new approach for constructing equilibrium measures which is pure geometrical in its nature and uses no symbolic representations of the system. As a result it can be used to effect thermodynamics formalism for systems for which no symbolic representation is available such as partially hyperbolic systems. This approach applies to a broad class of potentials satisfying Bowen’s property, which includes the usual class of Holder continuous potentials and it gives a new way for constructing measures of maximal entropy (first constructed in this setting by Margulis). It also reveals a crucial geometric property of equilibrium measures that has not been known before — the conditional measures they generate on unstable leaves are measures of full Caratheodory dimension — the fact that lies in the heart of the geometric approach. The talk is based on two joint work with V. Climenhaga and A. Zelerovich.