AI 中文总结
本文转载专著卷首内容，界定了数理等领域普遍存在的 Kantorovich 算子，发展其遍历理论，将 Aubry-Mather 理论扩展至哈密顿动力学之外的场景。
AI 中文摘要
本文转载了作者一部专著的前言、概述和目录等卷首内容，该专著以《斜线性熵与 Kantorovich 算子：一种广义的 Aubry-Mather 理论》为题提交出版。本书界定了一类非线性算子，我们将其命名为 Kantorovich 算子，这类算子在分析学、概率论、动力系统、数理经济学及金融学中普遍存在。我们以扩展涉及马尔可夫算子、自由能转移或 Hopf-Lax-Oleinik 半群的经典内容的方式，发展了这类算子的遍历理论相关内容。由于这类算子没有伴随算子，其与测度的对偶性通过概率分布对（源分布与目标分布）上的凸泛函实现，我们将该泛函命名为斜线性熵，它是最优质量输运的一般形式。此处转载的详尽概述描述了由此产生的遍历理论，其中极小测度、Mather 常数、弱 KAM 解及 Aubry 集被关联到任意一个 Kantorovich 算子，将 Aubry-Mather 理论的应用范围远远扩展到其在哈密顿动力学中的起源之外。
英文摘要
This paper reproduces the front matter --- preface, overview and table of contents --- of a monograph by the author, submitted for publication under the title {\it Skew Linear Entropies and Kantorovich Operators: A General Aubry-Mather Theory}. The book isolates a class of non-linear operators, which we call {\it Kantorovich operators}, that are ubiquitous in analysis, probability, dynamical systems, mathematical economics and finance. We develop aspects of their ergodic theory in a way that extends classical ones involving Markov operators, free-energy transfers, or the Hopf--Lax--Oleinik semi-group. Having no adjoint, the duality between such an operator and measures is carried instead via a convex functional on {\it pairs} of probability distributions --- a source and a target --- which we call a {\it skew-linear entropy}, and which is a general form of optimal mass transport. The extensive overview reproduced here describes the resulting ergodic theory, in which minimal measures, a Mather constant, weak KAM solutions and an Aubry set are attached to an arbitrary Kantorovich operator, extending Aubry--Mather theory well beyond its origins in Hamiltonian dynamics.
Comments33 pages, Updated version - if any - can be downloaded at https://www.birs.ca/~nassif/
