On the existence of a new type of periodic and quasi-periodic orbits for twist maps of the torus

Abstract

We prove that for a large and important class of C1 twist maps of the torus periodic and quasi-periodic orbits of a new type exist, provided that there are no rotational invariant circles (RICs). These orbits have a non-zero `vertical rotation number' (VRN), in contrast to what happens to Birkhoff periodic orbits and Aubry–Mather sets. The VRN is rational for a periodic orbit and irrational for a quasi-periodic. We also prove that the existence of an orbit with a VRN = a>0, implies the existence of orbits with VRN = b, for all 0<b<a. In this way, related to a generalized definition of rotation number, we characterize all kinds of periodic and quasi-periodic orbits a twist map of the torus can have. As a consequence of the previous results we obtain that a twist map of the torus with no RICs has positive topological entropy, which is a very classical result. At the end of the paper we present some examples, like the standard map, such that our results apply.