Denjoy subsystems and horseshoes

Summary

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.

All the dynamicists know the famous Poincaré sentence about periodic orbits:

Ce qui nous rend ces solutions périodiques si précieuses, c’est qu’elles sont, pour ainsi dire, la seule brèche par où nous puissions essayer de pénétrer dans une place jusqu’ici réputée inabordable.

But a periodic orbit for a dynamical system f : X → X is simply a finite invariant subset and the dynamics restricted to this set cannot be very complicated. What is more interesting is the dynamics close to such a periodic orbit, that may give rise to various rich phenomena. For example, for a symplectic diffeomorphism of a surface, two kinds of restricted dynamics to invariant Cantor sets can exist close to the periodic orbits, that are: