A convex analysis approach to entropy functions, variational principles and equilibrium states

Abstract

The first aims of this work are to endorse the advent of finitely additive set functions as equilibrium states and the possibility to replace the metric entropy by an upper semi-continuous map associated to a general variational principle. More precisely, using methods from Convex Analysis, we construct for each generalized convex pressure function an upper semi-continuous entropy-like map (which, in the context of continuous transformations acting on a compact metric space and the topological pressure, turns out to be the upper semi-continuous envelope of the Kolmogorov-Sinai metric entropy), then establish a new abstract variational principle and prove that equilibrium states, possibly finitely additive, always exist. This conceptual approach provides a new insight on dynamical systems without a measure with maximal entropy, prompts the study of finitely additive ground states for non-uniformly hyperbolic maps and grants the existence of finitely additive Lyapunov equilibrium states for singular value potentials generated by linear cocycles over continuous maps. We further investigate several applications, including a new thermodynamic formalism for systems driven by finitely generated semigroup or countable sofic group actions. On the final pages of the manuscript we provide a list of open problems in a wide range of topics suggested by our main results.