Absolutely continuous conjugacies of Blaschke products. III
Let \(f\) be a Blaschke product in the unit disk \(\mathbb{D}\). Its Julia set is the entire unit circle \(S\) if and only if it is ergodic. Let \(\varphi\) denote an absolutely continuous homeomorphism of \(S\) onto itself. In an earlier paper the author showed that \(\varphi \circ f \circ \varphi^{- 1}\) can be a Blaschke product only in the trivial case that \(\varphi\) is a Möbius transformation. The present paper studies the case that \(f\) is not ergodic. It is shown that there always exists \(\varphi\) such that \(f^* = \varphi \circ f \circ \varphi^{-1}\) is a finite Blaschke product but \(f^* \neq Q \circ f \circ Q^{-1}\) on \(S\) for all Möbius transformations \(\varphi\) of \(\mathbb{D}\) onto itself. The same result holds if ``finite Blaschke product'' is replaced by ``inner functions''. The proof uses the fact that there is an inner function \(g\) fixing 0 and a Möbius transformation \(T\) such that \(g \circ f = T \circ g\).
scientific article; zbMATH DE number 569426