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摘要
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 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 Lip (1/2) norm of at least C. We also provide a geometric criterion for traces to be driven by Lip (1/2) functions, and show that examples such as the Hilbert space-filling curve and the Sierpinski gasket fall into this class
- 出版日期2012
