= Boundary extension of the inverse map for a continuous Loewner trace
For a continuous <Loewner trace> generating its hulls, the inverse <mapping-out function of a compact H-hull> $g_t^{-1}$ extends continuously to the closed <complex upper half-plane>. The finite trace is a <continuous map> image of an interval, and the <boundary> of its unbounded complementary component has <local connectedness>; the <Caratheodory boundary extension theorem> gives the extension. In particular $g_t^{-1}(U_t)=\gamma(t)$, while $g_t^{-1}$ maps the open <complex upper half-plane> into $D_t$. This is the deterministic boundary-extension input of https://annals.math.princeton.edu/wp-content/uploads/annals-v161-n2-p07.pdf[Rohde and Schramm, Theorem 4.1]. Under a <Conformal Markov property of SLE> restart, this relates <boundary> contacts of the mapped future to contacts with the old hull, without assuming the old trace is simple.
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