AI 中文总结
本文研究紧度量空间上带线性约束的遍历优化问题，分析了约束测度集非空性、最优解存在性与唯一性等性质，刻画对偶问题并给出实现结果，推广了动力系统与最优传输中的相关优化问题。
AI 中文摘要
设$T:X\to X$是紧度量空间上的连续映射，$\phi:X\to\mathbb{R}$是连续函数。遍历优化问题是在所有$T$-不变的Borel概率测度$\mu$构成的集合上，最大化积分$\int\phi\\,d\mu$。本文研究遍历优化问题的带约束版本。给定一个“约束集”$\mathcal{C}\subset C(X)$，令$M_\mathcal{C}(X,T)$是$X$上满足对所有$g\in\mathcal{C}$都有$\int g\\,d\mu=0$的$T$-不变Borel概率测度$\mu$构成的集合。我们研究在约束集$M_\mathcal{C}(X,T)$上最大化积分$\int\phi\\,d\mu$的问题。我们探讨了该优化问题的基本性质，首先是$M_\mathcal{C}(X,T)$的非空性以及最优解的存在性。此外，我们证明了最优测度的通有唯一性和普遍唯一性，给出了一个实现结果，并刻画了对偶问题。该框架统一推广了此前动力系统和最优传输领域中若干已被研究的优化问题。
英文摘要
Let $T : X \to X$ be a continuous map of a compact metrizable space, and let $ϕ: X \to \mathbb{R}$ be a continuous function. The ergodic optimization problem is to maximize the integral $\int ϕ\, dμ$ 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' $\mathcal{C}\subset C(X)$, let $M_\mathcal{C}(X,T)$ be the set of $T$-invariant Borel probability measures $μ$ on $X$ such that $\int g \, dμ= 0$ for all $g \in \mathcal{C}$. We investigate the problem of maximizing the integral $\int ϕ\, dμ$ over the constrained set $M_\mathcal{C}(X,T)$. We address basic properties of this optimization problem, beginning with nonemptiness of $M_\mathcal{C}(X,T)$ 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.
Comments27 pages, no figure
