Variational construction of positive entropy invariant measures of Lagrangian systems and Arnold diffusion

TL;DR

This paper introduces a novel variational method to construct positive entropy invariant measures in Lagrangian systems, enabling the analysis of complex diffusion phenomena without restrictive assumptions on perturbations or manifold intersections.

Contribution

It develops a new variational approach for invariant measures in Lagrangian systems, applicable without transversality or smallness constraints, and provides bounds on entropy and drift in unstable regions.

Findings

Existence of shadowing ergodic measures with prescribed invariant tori.

Bounds on topological entropy and drift acceleration in instability regions.

Application of gradient dynamics of the action to construct invariant measures.

Abstract

We develop a variational method for constructing positive entropy invariant measures of Lagrangian systems without assuming transversal intersections of stable and unstable manifolds, and without restrictions to the size of non-integrable perturbations. We apply it to a family of two and a half degrees of freedom a-priori unstable Lagrangians, and show that if we assume that there is no topological obstruction to diffusion (precisely formulated in terms of topological non-degeneracy of minima of the Peierl's barrier function), then there exists a vast family of "horsheshoes", such as "shadowing" ergodic positive entropy measures having precisely any closed set of invariant tori in its support. Furthermore, we give bounds on the topological entropy and the "drift acceleration" in any part of a region of instability in terms of a certain extremal value of the Fr\'{e}chet derivative of the action functional, generalizing the angle of splitting of separatrices. The method of construction is new, and relies on study of formally gradient dynamics of the action (coupled parabolic semilinear partial differential equations on unbounded domains). We apply recently developed techniques of precise control of the local evolution of energy (in this case the Lagrangian action), energy dissipation and flux. In Part II of the paper we will apply the theory to obtain sharp bounds for topological entropy and drift acceleration for the same class of equations in the case of small perturbations.