4 problems

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 fo…

Brezis's conjecture. There exist constants $C>0$ and $\delta_0>0$ such that the displayed inequality holds for every continuous $f\in C^0({\mathbb S}^1;{\mathbb S}^1)$ and every…

Rigidity conjecture. If $f$ and $g$ have the same signature, then they are conjugate by a $C^1$ diffeomorphism. Moreover, if their common rotation number is of bounded type, then t…

Geometric rigidity conjecture. Any two $C^3$ critical circle maps with the same irrational rotation number of bounded type and the same odd criticality are conjugate to each other…