Ergodic optimization in dynamical systems

OLIVER JENKINSON

doi:10.1017/etds.2017.142

Abstract

Ergodic optimization is the study of problems relating to maximizing orbits and 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 any other invariant probability measure. In this paper, 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.

Highlights

For a real-valued function defined on the state space of a dynamical system, the topic of ergodic optimization revolves around understanding its largest possible ergodic average
Taking the dynamical system to be a map T : X → X, and denoting the function by f : X → R, attention is focused on the supremum of time averages limn→∞ (1/n) n−1 i =0 f (T i x) over those x for which the limit exists, or alternatively on the supremum of space averages f dμ over probability measures μ which are invariant under T
Sturmian measures, turned out to be unexpectedly ubiquitous in a variety of ergodic optimization problems, encompassing similar low-dimensional function spaces, certain infinite dimensional cones of functions, and problems concerning the joint spectral radius of matrix pairs

Summary

Introduction

For a real-valued function defined on the state space of a dynamical system, the topic of ergodic optimization revolves around understanding its largest possible ergodic average. For the case of (X, T ) a countable alphabet subshift of finite type, where X is non-compact and the entropy map μ → h(μ) is not upper semi-continuous, additional summability and boundedness hypotheses on the locally Holder function f : X → R, together with primitivity assumptions on X , ensure the existence and uniqueness of the equilibrium measures mt f , that the family (mt f ) does have an accumulation point m, and that h(m) = limt→∞ h(mt f ) = max{h(μ) : μ ∈ Mmax( f )} (see [63, 93, 124]), representing an analogue of Theorem 4.1. We note that zero temperature limits of equilibrium measures have been studied in a variety of other dynamical settings, including Frenkel–Kontorova models [6], quadratic-like holomorphic maps [54], multimodal interval maps [79] and Henon-like maps [152]

Revelations

Sturmian optimization and ergodic dominance