The behavior of renormalizations of circle maps with rational rotation numbers and with Zygmund conditions

Abstract

Let be one-parameter family of circle homeomorphisms with a break point, that is, the derivative has jump discontinuity at this point. Suppose satisfies a certain Zygmund condition which is dependent on parameter . We prove that the renormalizations of circle homeomorphisms from this family with rational rotation number of sufficiently large rank are approximated by piecewise fractional linear transformations in and -norms, depending on the values of the parameter and , respectively.