Rational rotation numbers for maps of the circle

TLDR: In this article, the authors consider families of maps of the circle of degree 1 which are homeomorphisms but not diffeomorphisms, and prove that the set of parameter values corresponding to irrational rotation numbers has Lebesgue measure 0.

Abstract: We consider families of maps of the circle of degree 1 which are homeomorphisms but not diffeomorphisms, that is maps like $$x \to x + t + \frac{c}{{2\pi }}\sin (2\pi x)(\bmod 1)$$ withc=1. We prove that the set of parameter values corresponding to irrational rotation numbers has Lebesgue measure 0. In other words, the intervals on which frequency-locking occurs fill up the set of full measure.