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

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