Renormalizations and rigidity theory for circle homeomorphisms with singularities of the break type

Circle homeomorphisms with singularities of the break type are considered in the case when rotation numbers have periodic continued fraction expansion. We establish hyperbolicity for renormalizations and then use it in order to prove the following rigidity result. Namely, we show that any two homeomorphisms with a single break point are smoothly conjugate to each other provided they have the same quadratic irrational rotation number and the same "size" of a break.