Rigidity of holomorphic Collet-Eckmann repellers

We prove rigidity results for a class of non-uniformly hyperbolic holomorphic maps. If a holomorphic Collet-Eckmann mapf is topologically conjugate to a holomorphic mapg, then the conjugacy can be improved to be quasiconformal. If there is only one critical point in the repeller, theng is Collet-Eckmann, too. The first author acknowledges support by Polish KBN Grant 2 P03A 025 12 “Iterations of Holomorphic Functions” and support of the Hebrew University of Jerusalem, where a part of tha paper was written. The second author is grateful for the hospitality and support of the Caltech, where a part of the paper was written.