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arXiv · 1103.0071

Spacefilling Curves and Phases of the Loewner Equation

Joan Lind·Steffen Rohde
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Abstract

Similar to the well-known phases of SLE, the Loewner differential equation with Lip(1/2) driving terms is known to have a phase transition at norm 4, when traces change from simple to non-simple 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 Lip(1/2) norm at least C. We also provide a geometric criterion for traces to be driven by Lip(1/2) functions, and show that for instance the Hilbert space filling curve and the Sierpinski gasket fall into this class.

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BibTeXRIS

Joan Lind, Steffen Rohde. 2011-03-01. Spacefilling Curves and Phases of the Loewner Equation. https://arxiv.org/abs/1103.0071

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