Let J be the set of inner functions whose derivative lies in Nevanlinna class. In this note, we show that the natural map F → Inn(F ) : J / Aut(D) → Inn /S 1 is injective but not surjective. More precisely, we show that that the image consists of all inner functions of the form BS µ where B is a Blaschke product and S µ is the singular factor associated to a measure µ whose support is contained in a countable union of Beurling-Carleson sets. Our proof is based on extending the work of D. Kraus and O. Roth on maximal Blaschke products to allow for singular factors. This answers a question raised by K. Dyakonov.

We study the interplay between infinitesimal deformations of conformal mappings, quasiconformal distortion estimates and integral means spectra. By the work of McMullen, the second derivative of the Hausdorff dimension of the boundary of the image domain is naturally related to asymptotic variance of the Beurling transform. In view of a theorem of Smirnov which states that the dimension of a k-quasicircle is at most 1 + k 2 , it is natural to expect that the maximum asymptotic variance Σ 2 = 1. In this paper, we prove 0.87913 Σ 2 1.For the lower bound, we give examples of polynomial Julia sets which are k-quasicircles with dimensions 1 + 0.87913 k 2 for k small, thereby showing that Σ 2 0.87913. The key ingredient in this construction is a good estimate for the distortion k, which is better than the one given by a straightforward use of the λ-lemma in the appropriate parameter space. Finally, we develop a new fractal approximation scheme for evaluating Σ 2 in terms of nearly circular polynomial Julia sets.

Abstract. We examine several characteristics of conformal maps that resemble the variance of a Gaussian: asymptotic variance, the constant in Makarov's law of iterated logarithm and the second derivative of the integral means spectrum at the origin. While these quantities need not be equal in general, they agree for domains whose boundaries are regular fractals such as Julia sets or limit sets of quasi-Fuchsian groups. We give a new proof of these dynamical equalities. We also show that these characteristics have the same universal bounds and prove a central limit theorem for extremals. Our method is based on analyzing the local variance of dyadic martingales associated to Bloch functions.