IUMJ

Title: Spacefilling curves and phases of the Loewner equation

Authors: Joan Lind and Steffen Rohde

Issue: Volume 61 (2012), Issue 6, 2231-2249

Abstract:

Similar to the well-known phases of SLE, the Loewner differential equation with $\operatorname{Lip}(1/2)$ driving terms is known to have a phase transition at norm $4$, when traces change from simple to nonsimple curves. We establish the deterministic analog of the second phase transition of SLE, where traces change to space-filling curves: there is a constant $C>4$ such that a Loewner driving term whose trace is space filling has $\operatorname{Lip}(1/2)$ norm of at least $C$. We also provide a geometric criterion for traces to be driven by $\operatorname{Lip}(1/2)$ functions, and show how examples such as the Hilbert space-filling curve and the Sierpinski gasket fall into this class.