  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.
