Vol. 10, No. 1, 2017

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Loewner deformations driven by the Weierstrass function

Joan Lind and Jessica Robins

Vol. 10 (2017), No. 1, 151–164

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.

Loewner evolution, Weierstrass function
Mathematical Subject Classification 2010
Primary: 30C35
Received: 15 September 2015
Accepted: 13 December 2015
Published: 11 October 2016

Communicated by Michael Dorff
Joan Lind
Department of Mathematics
University of Tennessee
Knoxville, TN 37996
United States
Jessica Robins
University of Tennessee
Knoxville, TN 37996
United States