Tropical thermodynamic formalism

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 $\mathcal{L}_A^{*}$ 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 $\mathcal{L}_A^{*}$ corresponding to $\mathcal{R}_A^{*}$, and establish the existence and generic uniqueness of tropical eigen-densities of $\mathcal{L}_A^{*}$. The Aubry set and the Ma\~{n}\'{e} 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 H\"{o}lder potential and establish a characterization theorem for the large deviation principle.