On Dilatations and Substantial Boundary Points of Homeomorphisms of Jordan Curves

We study the relation between the dilatations Kh and Kh* of a homeomorphism h of Jordan curves. We show that if Kh= Kh*, then either h is induced by an affine map or there is a substantial boundary point for h. In particular, we prove that if h is symmetric (in the sense of Gardiner and Sullivan), then Kh* > Kh. This is quite contrary to a previously conjectured relation between Kh and Kh*. © 1997, Birkhäuser Verlag, Basel.