Ergodic optimization in dynamical systems

TL;DR

This survey explores ergodic optimization in dynamical systems, focusing on maximizing orbits and measures, and discusses key models, properties, and classes of functions related to maximizing measures.

Contribution

It provides a comprehensive overview of ergodic optimization, including models, properties, and the zero temperature limit interpretation within thermodynamic formalism.

Findings

Maximizing measures often have Sturmian structure.

Adding coboundaries helps analyze properties of maximizing measures.

Typical properties vary across different function spaces.

Abstract

Ergodic optimization is the study of problems relating to maximizing orbits, maximizing invariant measures and maximum ergodic averages. An orbit of a dynamical system is called f-maximizing if the time average of the real-valued function f along the orbit is larger than along all other orbits, and an invariant probability measure is called f-maximizing if it gives f a larger space average than does any other invariant probability measure. In this survey we consider the main strands of ergodic optimization, beginning with an influential model problem, and the interpretation of ergodic optimization as the zero temperature limit of thermodynamic formalism. We describe typical properties of maximizing measures for various spaces of functions, the key tool of adding a coboundary so as to reveal properties of these measures, as well as certain classes of functions where the maximizing measure is known to be Sturmian.