A General Aubry-Mather Theory

Abstract

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.