On Makarov's principle in conformal mapping

TL;DR

This paper investigates various characteristics of conformal maps related to variance and fractal boundaries, establishing their equality in certain cases, providing new proofs, bounds, and a central limit theorem using martingale analysis.

Contribution

It introduces a new proof of dynamical equalities for conformal maps with fractal boundaries and demonstrates their universal bounds and a central limit theorem.

Findings

Equalities hold for Julia sets and limit sets of quasi-Fuchsian groups.

All characteristics share the same universal bounds.

A central limit theorem for extremals is established.

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.