# Journal of Operator Theory

Volume 50, Issue 2, Fall 2003 pp. 331-343.

A mapping theorem for the boundary set $X_T$ for an absolutely continuous contraction $T$**Authors**: G. Cassier, (1) I. Chalendar (2) and B. Chevreau (3)

**Author institution:**(1) Institut Girard Desargues, UFR de Mathematiques, Universite Claude Bernard Lyon, 69622 Villeurbanne Cedex, France

(2) Institut Girard Desargues, UFR de Mathematiques, Universite Claude Bernard Lyon, 69622 Villeurbanne Cedex, France

(3) Departement de Mathematiques, Universite Bordeaux I, 351 Cours de la Liberation, 33405 Talence Cedex, France

**Summary:**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 ${\mathbb T}$, of a sequence condition whose validity on all of ${\mathbb T}$ is equivalent to membership of $T$ in the class${\mathbb A}_{\aleph_0}$. The main result is the following: if $b$ is a Blaschke product of degree $d$ for which there exist $d$ distinct M\"obius transforms $u$ such that $b\circ u=b$, then $b(X_T)=X_{b(T)}$.

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