A Convex Analysis Approach to Entropy Functions, Variational Principles and Equilibrium States

Abstract

Using methods from Convex Analysis, for each generalized pressure function we define an upper semi-continuous affine entropy-like map, establish an abstract variational principle for both countably and finitely additive probability measures and prove that equilibrium states always exist. We show that this conceptual approach imparts a new insight on dynamical systems without a measure with maximal entropy, may be used to detect second-order phase transitions, prompts the study of finitely additive ground states for non-uniformly hyperbolic transformations and grants the existence of finitely additive Lyapunov equilibrium states for singular value potentials generated by linear cocycles over continuous self-maps.