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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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