Tropical thermodynamic formalism

TL;DR

This paper explores the tropical algebraic structure emerging from zero-temperature large deviation principles in dynamical systems, extending tropical functional analysis and establishing conditions for large deviation principles.

Contribution

It introduces the tropical adjoint Bousch operator, extends tropical functional analysis, and provides conditions for large deviation principles in the context of distance-expanding maps.

Findings

Existence and generic uniqueness of tropical eigen-densities established.

A sufficient condition for the large deviation principle is derived.

Characterization theorem for the large deviation principle is proven.

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 $\mathcal{L}_A^{*}$ 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.