Loewner deformations driven by the Weierstrass function

Lind, Joan; Robins, Jessica

doi:10.2140/involve.2017.10.151

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Loewner deformations driven by the Weierstrass function

Joan Lind and Jessica Robins

Vol. 10 (2017), No. 1, 151–164

DOI: 10.2140/involve.2017.10.151

Abstract

The Loewner differential equation provides a way of encoding growing families of sets into continuous real-valued functions. Most famously, Schramm–Loewner evolution (SLE) consists of the growing random families of sets that are encoded via the Loewner equation by a multiple of Brownian motion. The purpose of this paper is to study the families of sets encoded by a multiple of the Weierstrass function, which is a deterministic analog of Brownian motion. We prove that there is a phase transition in this setting, just as there is in the SLE setting.

Keywords

Loewner evolution, Weierstrass function

Mathematical Subject Classification 2010

Primary: 30C35

Milestones

Received: 15 September 2015

Accepted: 13 December 2015

Published: 11 October 2016

Communicated by Michael Dorff

Authors

Joan Lind
Department of Mathematics University of Tennessee Knoxville, TN 37996 United States

Jessica Robins
University of Tennessee Knoxville, TN 37996 United States

Vol. 10, No. 1, 2017