BOUNDARIES OF INSTABILITY ZONES FOR SYMPLECTIC TWIST MAPS

作者：Arnaud M C^*

来源：Journal of the Institute of Mathematics of Jussieu, 2014, 13(1): 19-41.

DOI：10.1017/S1474748013000029

• twist maps
• instability zones
• invariant curves
• irregularity of invariant curves
• Denjoy counter-examples
• connecting lemma
• Primary 37E40
• Secondary 37E10

Very few things are known about the curves that are at the boundary of the instability zones of symplectic twist maps. It is known that in general they have an irrational rotation number and that they cannot be KAM curves. We address the following questions. Can they be very smooth? Can they be non-C-1?
Can they have a Diophantine or a Liouville rotation number? We give a partial answer for C-1 and C-2 twist maps.
In Theorem 1, we construct a C-2 symplectic twist map f of the annulus that has an essential invariant Gamma curve such that
Gamma is not differentiable;
Gamma the dynamics of f(vertical bar Gamma) is conjugated to the one of a D enjoy counter-example;
Gamma is at the boundary of an instability zone for f.
Using the Hayashi connecting lemma, we prove in Theroem 2 that any symplectic twist map restricted to an essential invariant curve can be embedded as the dynamics along a boundary of an instability zone for some C-1 symplectic twist map.

• 出版日期2014-1