Connecting orbits for families of Tonelli Hamiltonians

TL;DR

This paper studies the existence of complex diffusion orbits in systems formed by iterating multiple Tonelli Hamiltonian maps, using weak KAM theory to extend results known for single Hamiltonian systems.

Contribution

It introduces a novel approach to analyze Arnold diffusion in polysystems of Tonelli Hamiltonians using weak KAM theory, generalizing previous results for twist maps.

Findings

Existence of Arnold diffusion-type orbits in polysystems.

Extension of known results from twist maps to more general Hamiltonian families.

Application of weak KAM theory to complex dynamical systems.

Abstract

We investigate the existence of Arnold diffusion-type orbits for systems obtained by iterating in any order the time-one maps of a family of Tonelli Hamiltonians. Such systems are known as 'polysystems' or 'iterated function systems'. When specialized to families of twist maps on the cylinder, our results are similar to those obtained by Moeckel [20] and Le Calvez [15]. Our approach is based on weak KAM theory and is close to the one used by Bernard in [3] to study the case of a single Tonelli Hamiltonian.