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The straightening theorem for tangent-like maps. (English) Zbl 1250.37025
Summary: By analogy to polynomial-like maps, we introduce a notion of tangent-like maps. The main result of this paper is the straightening theorem. It says that a tangent-like map is quasiconformally equivalent to some tangent-type function \(f : \mathbb C \rightarrow\bar\mathbb C \setminus {a,b}\) for \(a\neq b\), which is unique up to an affine map. We also prove that quasiconformal conjugacy is conformal on the interior of the filled Julia set.

MSC:
37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
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