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 $\kappa \in (0,4)$.

25 pages. Added a remark after Theorem 6.6; added Corollary B.5

References in corpus (2)

• Duality of Chordal SLE
• Reversibility of Whole-Plane SLE

Cited by in corpus (1)

• Backward SLE and the symmetry of the welding