Ergodic optimization, zero temperature limits and the max-plus algebra

TL;DR

This paper reviews ergodic theory, thermodynamic formalism, and max-plus algebra to explore properties of equilibrium measures as temperature approaches zero, highlighting their mathematical structure and behavior.

Contribution

It introduces the interplay between ergodic optimization, zero temperature limits, and max-plus algebra, providing new insights into equilibrium measures in this context.

Findings

Properties of equilibrium measures at zero temperature

Connections between ergodic theory and max-plus algebra

Behavior of thermodynamic formalism in the zero temperature limit

Abstract

Lecture notes of a course at the Brazilian Mathematical Colloquium. We review some basic notions in ergodic theory and thermodynamic formalism, as well as introductory results in the context of max-plus algebra, in order to exhibit some properties of equilibrium measures when temperature goes to zero.