Collisions and Spirals of Loewner Traces

We analyze Loewner traces driven by functions asymptotic to κ √ 1− t. We prove a stability result when κ 6= 4 and show that κ = 4 can lead to non locally connected hulls. As a consequence, we obtain a driving term λ(t) so that the hulls driven by κλ(t) are generated by a continuous curve for all κ > 0 with κ 6= 4 but not when κ = 4, so that the space of driving terms with continuous traces is not convex. As a byproduct, we obtain an explicit construction of the traces driven by κ √ 1− t and a conceptual proof of the corresponding results of Kager, Nienhuis and Kadanoff.