Tropical thermodynamic formalism

Abstract

We investigate the zero-temperature large deviation principle for equilibrium states in the context of distance-expanding maps. The logarithmic-type zero-temperature limit in the large deviation principle induces a tropical algebra structure, which motivates our study of the tropical adjoint Bousch operator ${\mathcal{L}}_A^{\ast }$ since the Bousch operator ${\mathcal{L}}_A$ is tropical linear and corresponds to the Ruelle operator ${\mathcal{R}}_A$. We extend tropical functional analysis, define the adjoint operator ${\mathcal{L}}_A^{\ast }$ corresponding to ${\mathcal{R}}_A^{\ast }$, and establish the existence and generic uniqueness of tropical eigen-densities of ${\mathcal{L}}_A^{\ast }$. The Aubry set and the Mañé potential, both originating from weak KAM theory, serve as important tools in the representation of tropical eigen-densities. We derive a sufficient condition for the large deviation principle which holds for a generic Hölder potential and establish a characterization theorem for the large deviation principle.

Some restructuring and polishing has been done to the current version from the previous one