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