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Rigidity of holomorphic Collet-Eckmann repellers. (English) Zbl 1034.37026
Let $$X$$ be a compact subset of the Riemann sphere $$P_1( {\mathbb C})$$, let $$U$$ be a neighborhood of $$X$$ and consider a holomorphic map $$f: U \to P_1({\mathbb C})$$ with $$f(X) =X$$. The pair $$(X,f)$$ is called a holomorphic repeller if the forward orbits of only the points of $$X$$ remain in a neighborhood of $$X$$. Following a definition introduced by P. Collet and J.-P. Eckmann about nonuniformly hyperbolic one-dimensional maps, the authors call a holomorphic repeller $$(X,f)$$ Coller-Eckmann if there are constants $$C>0$$ and $$\lambda > 1$$ such that for every critical point $$c\in X$$ of $$f$$ whose forward orbit does not meet other critical points of $$f$$, one has the estimate $$| (f^n)' (f(c))| \geq C \lambda^n$$ for all integers $$n\geq 0$$. Also, the authors call a holomorphic repeller $$(X,f)$$ topological Collet-Eckmann if there exists a constant $$d\geq 1$$ and for each $$x\in X$$ a set $$G(x)$$ of positive integers of lower density $$\geq { 1\over 2}$$ such that for every $$n\in G(x)$$, there exists a connected neighborhood of $$x$$ which is mapped properly by $$f^n$$ to a large disc, with topological degree at most $$d$$.
Their first theorem extends a result due to C. T. McMullen and D. Sullivan for rational maps [Adv. Math. 135, 351–395 (1998; Zbl 0926.30028)]: If $$(X,f)$$ and $$(Y,g)$$ are holomorphic repellers which are conjugate by an orientation-preserving homeomorphism $$h_0$$ and if $$(X,f)$$ is topological Collet-Eckmann, then there exists a semi-local quasiconformal conjugacy $$h$$ between $$f$$ and $$g$$ extending the restriction $$h_0|_X$$ to a neighborhood of $$X$$. In the proof, they apply the classical characterization of quasiconformal mappings due to J. Heinonen and P. Koskela [Invent. Math. 120, 61–79 (1995; Zbl 0832.30013)]. As one of the corollaries (also obtained independently by P. W. Jones and S. Smirnov [Ark. Mat. 38, 263–279 (2000)]) they deduce that if $$f$$ and $$g$$ are holomorphic polynomials whose filled-in Julia sets are connected and equal to their Julia sets, which are topologically conjugate by an orientation-preserving homeomorphism and if $$(J(f),f)$$ is Collet-Eckmann (or more generally, topological Collet-Eckmann), then $$f$$ and $$g$$ are conjugate by a conformal affine map.
Other interesting results are derived. Their proof relies on the method of “shrinking of neighborhoods” introduced by the first author [Trans. Am. Math. Soc. 350, 717–742 (1998; Zbl 0892.58063)] in order to control distortion as well as J. Graczyk and S. Smirnorv’s [Invent. Math. 133 , 69–96 (1998; Zbl 0916.30023)] reversed telescopic construction.

##### MSC:
 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets 37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 30C65 Quasiconformal mappings in $$\mathbb{R}^n$$, other generalizations 37C70 Attractors and repellers of smooth dynamical systems and their topological structure 37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
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