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The first aims of this work are to endorse the advent of finitely additiveset functions as equilibrium states and the possibility to replace the metricentropy by an upper semi-continuous map associated to a general variationalprinciple. More precisely, using methods from Convex Analysis, we construct foreach generalized convex pressure function an upper semi-continuous entropy-likemap (which, in the context of continuous transformations acting on a compactmetric space and the topological pressure, turns out to be the uppersemi-continuous envelope of the Kolmogorov-Sinai metric entropy), thenestablish a new abstract variational principle and prove that equilibriumstates, possibly finitely additive, always exist. This conceptual approachprovides a new insight on dynamical systems without a measure with maximalentropy, prompts the study of finitely additive ground states for non-uniformlyhyperbolic maps and grants the existence of finitely additive Lyapunovequilibrium states for singular value potentials generated by linear cocyclesover continuous maps. We further investigate several applications, including anew thermodynamic formalism for systems driven by finitely generated semigroupor countable sofic group actions. On the final pages of the manuscript weprovide a list of open problems in a wide range of topics suggested by our mainresults.

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