Elliptic stars in a chaotic night

A recurrent theme in the description of phase portraits of dynamical systems is that of elliptic islands in a chaotic sea. Usually this picture is invoked in the context of smooth twist maps of the annulus or the torus, like the standard map. In this setting ‘elliptic islands’ refers to the topological disks bounded by periodic smooth curves surrounding elliptic periodic points. Establishing the existence of these curves is one of the many achievements of KAM-theory. The aim of this note is to approach the topic from a different angle, namely from the viewpoint of rotation theory in a purely topological setting. We study homeomorphisms of the two-torus, homotopic to the identity, which have no wandering open sets (as in the area-preserving case) and whose rotation set has non-empty interior. We define local rotation subsets ρF (U) by restricting Misiurewicz and Ziemian’s definition of the rotation set to starting points in a small open disk U . Our main result is the following dichotomy: Either ρU (F ) is reduced to a single rational vector and U is contained in a periodic topological open disk which contains a periodic point, or ρU (F ) is large, in the sense that its convex hull has non-empty interior. This allows to distinguish an ‘elliptic’ and a ‘chaotic’ regime, and as a consequence we obtain that in the chaotic region the dynamics are sensitive with respect to initial conditions. In order to demonstrate these results we introduce a parameter family of smooth toral diffeomorphisms that is inspired by an example of Misiurewicz and Ziemian. The pictures obtained from simulations in this family motivate an alternative formulation of the original theme. Figure 1: Elliptic islands surrounding two 2-periodic orbits of the map fα(x, y) = (x+α sin(2π(y+ α sin(2πx))), y + α sin(2πx)) with α = 0.5. ∗TU Dresden, Institut für Analysis. Email: [email protected]