Idempotent approach to level-2 variational principles in thermodynamical formalism

In this work, we adapt the idempotent formalism to the level-2 setting, introducing the concept of idempotent pressure and its associated density entropy at level-2. In a broad sense, level-2 corresponds to the study of potentials or probabilities defined on the set of probabilities. The idempotent pressure is a natural concept that corresponds to the meaning of measure in the level-2 Max-Plus context. Given an idempotent pressure and a potential g, we investigate the equilibrium states which maximize an associated variational principle akin to the topological pressure; they are level-1 probabilities and not necessarily unique. Our results can be seen as part of Tropical Geometry. We also study a level-2 max-plus iterated function systems operator, which acts on idempotent pressures and is defined from a family of dual Ruelle operators as the maps of the IFS acting on probabilities; requiring the control of a uniform bound of contraction for an uncountable family of classical dual-Ruelle operators. We prove the existence of a unique idempotent pressure which is fixed under its action. This idempotent pressure is also fixed by the action of the push-forward map.