Denjoy sub-systems and horseshoes

TL;DR

This paper introduces weak Denjoy subsystems as a generalization of Aubry-Mather sets, constructs a family of such subsystems within horseshoes linked to rotation numbers, and shows most Aubry-Mather sets in generic twist maps are contained in horseshoes.

Contribution

It defines weak Denjoy subsystems, associates rotation numbers to them, and demonstrates their presence in horseshoes and Aubry-Mather sets for generic twist maps.

Findings

Construction of a continuous family of WDS in horseshoes indexed by rotation number

Most Aubry-Mather sets are contained in horseshoes for generic twist maps

A rotation number can be associated to weak Denjoy subsystems

Abstract

We introduce a notion of weak Denjoy subsystem (WDS) that generalizes the Aubry-Mather Cantor sets to diffeomorphisms of manifolds. We explain how a rotation number can be associated to such a WDS. Then we build in any horseshoe a continuous one parameter family of such WDS that is indexed by its rotation number. Looking at the inverse problem in the setting of Aubry-Mather theory, we also prove that for a generic conservative twist map of the annulus, the majority of the Aubry-Mather sets are contained in some horseshoe that is associated to a Aubry-Mather set with a rational rotation number.