A Farey tree organization of locking regions for simple circle maps

Abstract

Let f f be a C 3 C^3 circle endomorphism of degree one with exactly two critical points and negative Schwarzian derivative. Assume that there is no real number a a such that f + a f + a has a unique rotation number equal to p q \frac {p}{q} . Then the same holds true for any p ′ q ′ \frac {p’}{q’} such that p q \frac {p}{q} stands above p ′ q ′ \frac {p’}{q’} in the Farey tree and can be related to it by a path on the tree.