Singular inner functions and associated measures (Q1346458)

Every inner function \(S\) without zeros in the unit disc \(D\) has a unique representation \(S(z) = \lambda \exp \{- \int_{|z |= 1} {e^{it} + z \over e^{it} - z} d \mu (t)\}\), where \(|\lambda |= 1\) and \(\mu\) is a finite positive measure singular with respect to Lebesgue measure. The authors find explicit characterisations (in terms of \(\mu)\) of the measure corresponding to \(\mu\) for the composition \(S \circ \varphi\) when \(\varphi\) is (a) an analytic endomorphism of \(\partial D\) or (b) a conformal map of \(D\) to itself.