Backward SLE and the symmetry of the welding

TL;DR

This paper demonstrates that backward SLE conformal weldings are symmetric under inversion and establishes their reversibility, using analytic and coupling techniques.

Contribution

It proves the invariance and reversibility of backward SLE weldings, revealing new symmetry properties in the theory of Schramm-Loewner Evolution.

Findings

Backward SLE weldings are invariant under x→-1/x.

The associated welding solutions are reversible.

The proofs involve analytic circle diffeomorphisms and coupling methods.

Abstract

The backward chordal Schramm-Loewner Evolution naturally defines a conformal welding homeomorphism of the real line. We show that this homeomorphism is invariant under the automorphism $x\mapsto -1/x$, and conclude that the associated solution to the welding problem (which is a natural renormalized limit of the finite time Loewner traces) is reversible. The proofs rely on an analysis of the action of analytic circle diffeomorphisms on the space of hulls, and on the coupling techniques of the second author.