| Rigidity of Holomorphic Collet-Eckmann Repellers (1997) | |||||||||||||||
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| . We prove rigidity results for a class of non-uniformly hyperbolic holomorphic maps: If a holomorphic Collet-Eckmann map f is topologically conjugate to a holomorphic map g, then the conjugacy can be improved to be quasiconformal. If there is only one critical point in the repeller, then g is Collet-Eckmann, too. 1. Introduction Collet-Eckmann maps of the interval were introduced by P. Collet and J.-P. Eckmann as a large class of non-uniformly expanding maps for which a probability absolutely continuous invariant measure exists. A theory of rational Collet-Eckmann maps was originated in [P2] and continued in [P3], [GS] and [PR]; see [PR] for a more detailed historical account. This paper is a continuation of [PR]. We consider repellers for holomorphic maps, without assuming the maps extend to rational maps. Consider a compact set X in the Riemann sphere C , together with a holomorphic map f : U ! C with f(X) = X, where U is a neighbourhood of X. We call the pair (X; f) a holomorp... | |||||||||||||||
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