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

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

2002-07-16

arxiv.org/abs/math/0207133v1

Mathematics

Dynamical Systems

20 pages. to appear in Nonlinearity 15(5) 1399-1416

Scientific paper

10.1088/0951-7715/15/5/303

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 $V.R.N=a>0,$ implies the existence of orbits with $V.R.N=b,$ for all $0

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