In the classical settings of Anosov diffeomorphisms or more general locally maximal hyperbolic sets I describe a new approach for constructing equilibrium measures corresponding to some continuous potentials and for studying some of their ergodic properties. This approach 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. Furthermore, this approach gives a new way for constructing measures of maximal entropy (first constructed in this setting by Margulis). It also reveals a crucial new geometric property of equilibrium measures — the conditional measures they generate on (un)stable leaves are measures of full Caratheodory dimension — the fact that lies in the heart of the geometric approach. The talk is based on a joint work with V. Climenhaga and A. Zelerovich.
Tue, 21/11/2017 - 14:15 to 15:15