Ergodic Optimization with Linear Constraints

2026/08/03 by Shengwen Guo, Kevin McGoff
Mathematics · #math.DS

paper · pdf

27 pages, no figure

arxiv created 2026/08/03 · arxiv updated 2026/08/04

Abstract

Let T : X → X be a continuous map of a compact metrizable space, and let ϕ: X → ℝ be a continuous function. The ergodic optimization problem is to maximize the integral ∫ ϕ dμ as μ ranges over all T-invariant Borel probability measures on X. In this paper we consider a constrained version of the ergodic optimization problem. Given a `constraint set' C⊂ C(X), let M_C(X,T) be the set of T-invariant Borel probability measures μ on X such that ∫ g dμ= 0 for all g ∈ C. We investigate the problem of maximizing the integral ∫ ϕ dμ over the constrained set M_C(X,T). We address basic properties of this optimization problem, beginning with nonemptiness of M_C(X,T) and existence of optimal solutions. Additionally, we establish the generic and prevalent uniqueness of optimal measures, we provide a realization result, and we give a characterization of the dual problem. This framework provides a common generalization of several previously considered optimization problems in dynamical systems and optimal transport.

Citations

• Typical Uniqueness in Ergodic Optimization
• Prevalent uniqueness in ergodic optimisation
• Ergodic optimization theory for a class of typical maps
• Ergodic optimization in dynamical systems
• Haar null and Haar meager sets: a survey and new results
• Variational analysis of inference from dynamical systems
• Relative equilibrium states and class degree
• On the Monge-Kantorovich problem with additional linear constraints
• Ground States are generically a Periodic Orbit
• Geometry and entropy of generalized rotation sets
• Functions for relative maximization
• Measures of maximal relative entropy