Inner Functions, Möbius Distortion and Angular Derivatives

Abstract: We prove that an inner function has finite $\mathcal{L} (p)$-entropy if and only if its accumulated M\"obius distortion is in $L^p$, $0<p<\infty$. We also study the support of the positive singular measures such that their corresponding singular inner functions have finite $\mathcal{L} (p)$-entropy.