Tropical thermodynamic formalism

We investigate the correspondence between thermodynamic formalism and ergodic optimization. It has been known that the Bousch operator LA is tropical linear and corresponds to the Ruelle operator RA. In this paper, we present general idempotent analysis (tropical algebra) results, define the tropical adjoint operator LA^ which corresponds to RA*, and study the existence and generic uniqueness of tropical eigen-densities of LA^. We also investigate the Logarithmic type zero temperature limit of equilibrium states which implies the large deviation principle. It turns out that the rate function is the tropical product of a calibrated sub-action and a tropical eigen-density.