Ergodicity of the tip of an SLE curve

Abstract

We first prove that, for $$\kappa \in (0,4)$$ , a whole-plane $$\hbox {SLE}(\kappa ;\kappa +2)$$ trace stopped at a fixed capacity time satisfies reversibility. We then use this reversibility result to prove that, for $$\kappa \in (0,4)$$ , a chordal $$\hbox {SLE}_\kappa $$ curve stopped at a fixed capacity time can be mapped conformally to the initial segment of a whole-plane $$\hbox {SLE}(\kappa ;\kappa +2)$$ trace. A similar but weaker result holds for radial $$\hbox {SLE}_\kappa $$ . These results are then used to study the ergodic behavior of an SLE curve near its tip point at a fixed capacity time. The proofs rely on the symmetry of backward SLE weldings and conformal removability of $$\hbox {SLE}_\kappa $$ curves for $$\kappa \in (0,4)$$ .