Greene’s noble-circle breakup conjecture

Determine whether the last invariant curves to break in a two-degree-of-freedom area-preserving map are those with noble rotation numbers, and, specifically, whether the golden-mean invariant circle is the final invariant circle to break in the standard map.

Background

The paper discusses nearly integrable Hamiltonian systems and the destruction of invariant tori under perturbation. For two-dimensional area-preserving maps, Greene’s residue criterion studies the stability of periodic orbits whose rotation numbers approximate a given irrational rotation number.

The conjecture concerns the ordering of torus destruction as the perturbation strength increases. Noble rotation numbers are defined by continued-fraction expansions ending in an infinite tail of ones; the golden mean is the principal example. The notes state that the criterion has only partial rigorous justification and is mainly used numerically to estimate the critical perturbation strength.

References

For d=2 one can ask which torus is the most robust. Greene's residue criterion, which tracks the stability of periodic orbits whose rotation numbers converge to a given W, led to the conjecture that the last curves to break are those with noble rotation number, a continued-fraction expansion ending in an infinite tail of ones, and that in the standard map the very last is the golden-mean circle W=(\sqrt5-1)/2. Which noble circle survives longest is map-dependent. The criterion remains a conjecture with only partial rigorous justification, but it is the standard numerical route to the critical perturbation strength.

— Classical and Quantum Chaos from Foundations to Holography  (2608.19131 - Shukla, 19 Aug 2026) in Section 2, subsection “Integrable tori and the KAM picture” (label sec:L2_kam)