Inner Functions, Möbius Distortion and Angular Derivatives

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