Rafael de la Llave, Arturo Olvera
The Obstruction criterion for non existence of Invariant Circles
and Renormalization.
(1062K, PostScript)
ABSTRACT. We formulate a conjecture which supplements the standard renormalization
scenario for the breakdown of golden circle in twist maps.
We show rigorously that if the conjecture was true then:
a) The stable manifold of the non-trivial fixed point would indeed be
a boundary between the existence of smooth invariant tori and hyperbolic
orbits with golden mean rotation number. In particular, the boundary of
the set of twist maps with a torus with a golden mean rotation number would
include a smooth submanifold in the space of analytic mappings.
b) The obstruction criterion of [Olvera-Simo] would be sharp in the
universality class of the renormalization group.
c) The criterion of [Greene-79] for existence of invariant circles if and
only if there the residues of approximating orbits are finite would be valid
for maps in the universality class.
d) If there is no invariant circle, there are hyperbolic sets with golden
mean rotation number.
We also provide numerical evidence which suggests that
the conjecture is true and discuss briefly the possibilities of providing a
computer-assisted proof. We also discuss tentatively phenomena
for other rotation numbers.