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Similar to the well-known phases of SLE, the Loewner differential equationwith Lip(1/2) driving terms is known to have a phase transition at norm 4, whentraces change from simple to non-simple curves. We establish the deterministicanalog of the second phase transition of SLE, where traces change tospace-filling curves: There is a constant C>4 such that a Loewner driving termwhose trace is space filling has Lip(1/2) norm at least C. We also provide ageometric criterion for traces to be driven by Lip(1/2) functions, and showthat for instance the Hilbert space filling curve and the Sierpinski gasketfall into this class.

Spacefilling Curves and Phases of the Loewner Equation