The obstruction criterion for non-existence of invariant circles and renormalization

The goal of this paper is to show that the renormalization group and the obstruction criterion can work together. We formulate a conjecture which supplements the standard renormalization scenario for the breakdown of golden circles in twist maps. We show rigorously that if the conjecture was true then the following hold.

1. The stable manifold of the non-trivial fixed point would be part of the 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. Moreover, if the conjecture was true, in the domain of universality (i.e. a small neighbourhood of the non-trivial fixed point), we would have the following (2), (3), (4).

2. The obstruction criterion for non-existence of golden mean invariant circles (Olvera and Simó 1987 Physica D 26 181–92) is sharp. That is, for maps in the universality class there is either a golden invariant circle or the condition in Olvera and Simó for non-existence of golden circles applies.

3. The criterion of Greene (1979 J. Math. Phys. 20 1183–201) for existence of invariant circles if and only if there the residues of approximating orbits are finite would be valid. That is, for maps in the universality class there would be a smooth invariant circle if and only if the residue of periodic orbits approximating the circle goes to zero.

4. If there is no invariant circle, there are uniformly hyperbolic sets with golden mean rotation number.

We also provide numerical evidence which suggests that the conjecture is true. We derive several scaling relations for observables related to the obstruction criterion and verify them.