Backward SLE and the symmetry of the welding

Understand

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.