On Makarov's principle in conformal mapping

Oleg Ivrii*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)1543-1567
Number of pages25
JournalInternational Mathematics Research Notices
Issue number5
StatePublished - 6 Mar 2019
Externally publishedYes


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