Rigidity for circle diffeomorphisms with singularities

This paper reviews recent results related to rigidity theory for circle diffeomorphisms with singularities. Both diffeomorphisms with a break point (sometimes called a ‘fracture-type singularity’ or ‘weak discontinuity’) and critical circle maps are discussed. In the case of breaks, results are presented on the global hyperbolicity of the renormalization operator; this property implies the existence of an attractor of the Smale horseshoe type. It is also shown that for maps with singularities rigidity is stronger than for diffeomorphisms, in the sense that rigidity is not violated for non-generic rotation numbers, which are abnormally well approximable by rationals. In the case of critical rotations of the circle it is proved that any two such rotations with the same order of the singular point and the same irrational rotation number are $C^1$-smoothly conjugate.