Renormalization conjecture and rigidity theory for circle diffeomorphisms with breaks

We prove the renormalization conjecture for circle diffeomorphisms with breaks, i.e., that the renormalizations of any two C (2+alpha) -smooth (alpha a (0, 1)) circle diffeomorphisms with a break point, with the same irrational rotation number and the same size of the break, approach each other exponentially fast in the C (2)-topology. As was shown in [KKM], this result implies the following strong rigidity statement: for almost all irrational numbers rho, any two circle diffeomorphisms with a break, with the same rotation number rho and the same size of the break, are C (1)-smoothly conjugate to each other. As we proved in [KK13], the latter claim cannot be extended to all irrational rotation numbers. These results can be considered an extension of Herman's theory on the linearization of circle diffeomorphisms.