Ergodicity of the tip of an SLE curve

Abstract

We first prove that, for $\kappa\in(0,4)$, a whole-plane 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 SLE$_\kappa$ curve stopped at a fixed capacity time can be mapped conformally to the initial segment of a whole-plane SLE$(\kappa;\kappa+2)$ trace. A similar but weaker result holds for radial 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 laminations and conformal removability of SLE$_\kappa$ curves for $\kappa\in(0,4)$.