Distortion and Distribution of Sets Under Inner Functions

Abstract

It is a classical result that Lebesgue measure on the unit circle is invariant under inner functions fixing the origin. In this setting, the distortion of Hausdorff contents has also been studied. We present here similar results focusing on inner functions with fixed points on the unit circle. In particular, our results yield information not only on the size of preimages of sets under inner functions, but also on their distribution with respect to a given boundary point. As an application, we use them to estimate the size of irregular points of inner functions omitting large sets. Finally, we also present a natural interpretation of the results in the upper half plane.

A Central Limit Theorem for Inner Functions

“…Several authors have also studied the distortion of Hausdorff measures by this mapping. See [FP92] and [LNS19]. Dynamical properties of the mapping f : BD Ñ BD, as recurrence, ergodicity, mixing, entropy and others have been studied by Aaranson [Aar78], Crazier [Cra91], Doering and Mañe [DM91], Fernández, Melián and Pestana [FMP07], [FMP12], Neurwirth [Neu78], Pommerenke [Pom81], and others.…”