Devil's-staircase conjecture for quasiperiodically forced Arnold circle maps

Let $\Lambda_\infty$ be the parameter set from Corollary

, and let $\tau=\frac{1}{2}$ denote the parameter in the corresponding situation. A mode-locking plateau is **non-collapsed** when it has nonzero parameter length. **Devil's-staircase conjecture.** The set $\Lambda_\infty$ can be chosen so that it is contained in the closure of the union of non-collapsed mode-locking plateaus. The same is true for the parameter $\tau=\frac{1}{2}$ in the situation of Corollary

.

This conjecture concerns whether the rotation number has a devil's-staircase structure for quasiperiodically forced Arnold circle maps. The text states that no rigorous results were known for this map and that numerical evidence was ambiguous, while suggesting that the techniques developed in the paper might prove the claim.