Notes on Tan ’ s theorem on similarity between the Mandelbrot set and the Julia sets ∗
نویسنده
چکیده
This note gives a simplified proof of the similarity between the Mandelbrot set and the quadratic Julia sets at the Misiurewicz parameters, originally due to Tan Lei [TL]. We also give an alternative proof of the global linearization theorem of repelling fixed points. 1 Simlarity between M and J We first give a proof of Tan’s theorem, in a way inspired by Schwick [Sch]. The Julia sets and the Mandelbrot set. Let us consider the quadratic family { fc(z) = z 2 + c : c ∈ C } . The Mandelbrot set M is defined by M := {c ∈ C : |f c (c)| ≤ 2 (∀n ∈ N)}. For each c ∈ C, the filled Julia set Kc is defined by Kc := {z ∈ C : |f c (z)| ≤ max{2, |c|} (∀n ∈ N)}. The Julia set Jc is the boundary of Kc. It is known that all M,Kc, and Jc (for any c ∈ C) are non-empty compact sets. Misiurewicz parameters. We say c0 ∈ ∂M is a Misiurewicz parameter if the forward orbit of c0 by fc0 eventually lands on a repelling periodic point. More precisely, there exist minimal l, p ≥ 1 such that a0 := f l c0(c0) satisfies a0 = f p c0 (a0) and |(f c0) (a0)| > 1. Note that the repelling periodic point a0 is stable: that is, there exists a neighborhood V of c0 with the following property: there exists a conformal map a : c 7→ a(c) on V such that a(c0) = a0; a(c) = f p c (a(c)); and |(f c )′(a(c))| > 1. In the following we set λ(c) := (f c ) ′(a(c)) and λ0 := λ(c0). A key lemma. This is our key lemma, which bridges the dynamical and paramter planes: ∗ver. 20130924.
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