A mapping theorem for the boundary set XT of an absolutely continuous contraction T

Let T be an absolutely continuous contraction acting on a Hilbert space. Its boundary set X-T can be seen as a localization, on a Borel subset of the unit circle T, of a sequence condition whose validity on all of T is equivalent to membership of T in the classA(N0). The main result is the following: if b is a Blaschke product of degree d for which there exist d distinct Mobius transforms u such that b omicron u = b, then b(X-T) = X-b(T).