Idempotent approach to level-2 variational principles in Thermodynamical Formalism

In this work we introduce an idempotent pressure to level-2 functions and its associated density entropy. All this is related to idempotent pressure functions which is the natural concept that corresponds to the meaning of probability in the level-2 max-plus context. In this general framework the equilibrium states, maximizing the variational principle, are not unique. We investigate the connections with the general convex pressure introduced recently to level-1 functions by Bi\'{s}, Carvalho, Mendes and Varandas. Our general setting contemplates the dynamical and not dynamical framework. We also study a characterization of the density entropy in order to get an idempotent pressure invariant by dynamical systems acting on probabilities; this is therefore a level-2 result. We are able to produce idempotent pressure functions at level-2 which are invariant by the dynamics of the pushforward map via a form of Ruelle operator.