Difference between revisions of "Perturbed Fatou coordinates"

Revision as of 20:27, 4 June 2011

I will use this (emerging) page to quote Shishikura, and use perhaps this article to draw some conclusions later.

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-Shishikura writes<ref name="Shishikura2000">Shishikura, M. (2000). Bifurcation of parabolic fixed points. In Lei, Tan, The Mandelbrot set, theme and variations. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 274, 325-363 (2000)</ref> in
+I will use [[Shishikura_perturbed_Fatou_coordinates|this (emerging) page]] to quote Shishikura, and use perhaps this article to draw some conclusions later.
-== The statements of Shishikura ==
-Shishikura writes<ref name="Shishikura2000">Shishikura, M. (2000). Bifurcation of parabolic fixed points. In Lei, Tan, The Mandelbrot set, theme and variations. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 274, 325-363 (2000)</ref> the following
-=== p. 327 ===
-If $f_0''(0)\neq 0$ by another coordinate change we may assume that $f_0''(0)=1$, so define
-$$ \mathcal{F}_0 = \{ f_0 \colon f_0 \quad\text{is defined and analytic in a neighborhood of 0}\quad f_0(0)=0, f_0'(0)=1, f_0''(0)=1\}$$
-=== p. 339 ===
-$$ \mathcal{F} = \{ f \colon f \quad\text{is defined and analytic in a neighborhood of 0}\quad f(0)=0, f_0'(0)\neq 0\}$$
-For $f\in \mathcal{F}$ we express the derivative
-$$f'(0)=\exp(2\pi i \alpha(f))$$
-where $\alpha(f)\in\C$ and $-\frac{1}{2} < \Re(\alpha(f)) \le \frac{1}{2}$ our study is focussed on the maps in the following class
-$$ \mathcal{F}_1 = \{ f\in \mathcal{F}\colon |\arg(\alpha(f))| < \frac{\pi}{4}\} $$
-=== Proposition 4.4.1 p. 356 ===
-Let $f_0\in\mathcal{F}_0$, there exists a neighborhood $\mathcal{N}_0$ of $\mathcal{F}_0$, such that if $f\in \mathcal{N}_0 \cap \mathcal{F}_1$, then there exist large $\xi_0, \eta_0>0$, and analytic maps $\ph_f\colon D(\ph_f)\to\overline{\C}$ and $\tilde{\mathcal{R}}_f\colon D(\tilde{\mathcal{R}}_f\to \C$ satisfying the following:
-# $D(\ph_f)$ contains the set $$ \mathcal{Q}_f = \{ w\in\C\colon \arg(-w-\xi_0) < \frac{2\pi}{3}, |\arg(w+\frac{1}{\alpha(f)}-\xi_0)|<\frac{2\pi}{3} \} $$ and $\ph_f(D(\ph_f)) \subseteq D(f)$; If $w, w+1\in D(\ph_f)$ then $$ \ph_f(w+1)=f(\ph_f(w)) $$ $\ph_f(w)\to 0$ when $w\in \mathcal{Q}_f$ and $\Im(w)\to\infty$; $\ph_f(w)\to\sigma(f)$ when $w\in\mathcal{Q}_f$ and $\Im(w)\to -\infty$.
-# $D(\tilde{\mathcal{R}}_f)$ contains $\{ w\in\C\colon \Im(w) > \eta_0\}$ and is invariant under $T(w)=w+1$; if $w\in D(\tilde{\mathcal{R}}_f)$ then $$\tilde{\mathcal{R}}_f(w+1)=\tilde{\mathcal{R}}(w)+1;$$ $\tilde{\mathcal{R}}_f(w)-w$ tends to $-\frac{1}{\alpha(f)}$ when $\Im(w)\to\infty$; to a constant when $\Im(w)\to -\infty$.
-# If $w\in D(\tilde{\mathcal{R}}_f\cap D(\ph_f)$ and $w'=\tilde{\mathcal{R}}_f(w)+n\in D(\ph_f)$ for some $n\in\Z$ then $$f^n(\ph_f(w))=\ph_f(w')\quad \text{for}\quad n\ge 0, \quad f^{-n}(\ph_f(w'))=\ph_f(w)\quad \text{for}\quad n<0$$ Moreover if $|\arg(w'+\frac{1}{2\alpha(f)}-\xi_0)|<\frac{2\pi}{3}$ then $n>0$.
-# When $f\in\mathcal{F_1}$ and $f\to f_0$ $$\ph_f\to\ph_0\quad\text{and}\quad \tilde{\mathcal{R}}_f + \frac{1}{\alpha(f)} \to \tilde{\mathcal{E}}_{f_0}$$
-<references/>

Difference between revisions of "Perturbed Fatou coordinates"

Revision as of 20:27, 4 June 2011

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