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

We prove that for a large and important class of $C^1$ twist maps of the torus periodic and quasi-periodic orbits of a new type exist, provided that there are no rotational invariant circles (R.I.C’s). These orbits have a non-zero ‘‘vertical rotation number’’ (V.R.N.), in contrast to what happens to Birkhoff periodic orbits and Aubry-Mather sets. The V.R.N. is rational for a periodic orbit and irrational for a quasi-periodic. We also prove that the existence of an orbit with a $V\mathrm{.}R\mathrm{.}N=a>0,$ implies the existence of orbits with $V\mathrm{.}R\mathrm{.}N=b,$ for all $0<b<a\mathrm{.}$ 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. And as a consequence of the previous results we obtain that a twist map of the torus with no R.I.C’s has positive topological entropy, which is a very classical result. In the end of the paper we present some examples, like the Standard map, such that our results apply.