Difference between revisions of "Perturbed Fatou coordinates"
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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 proposition 4.4.1 p. 356 | + | 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 |
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| + | == 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 | ||
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| + | === p. 327 === | ||
| + | If $f''_0(0)\neq 0$ by another coordinate change we may assume that $f_0''(0)=1$, so define | ||
| + | $$ \mathcal{F}_0 = \{ f_0 \colon f_0 \quad\text{is defined and analytic in a neighborhood of 0}\quad f_0(0)=0, f_0'(0)=1, f_0''(0)=1\}$$ | ||
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| + | === 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 | ||
| + | $$ \mathcal{F}_1 = \{ f\in \mathcal{F}\colon |\arg(\alpha(f))| < \frac{\pi}{4}\} $$ | ||
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| + | === 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 $f_0\in \mathcal{N}_0 \cap \mathcal{F}_1$, 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: | Let $f_0\in\mathcal{F}_0$, there exists a neighborhood $\mathcal{N}_0$ of $\mathcal{F}_0$, such that if $f_0\in \mathcal{N}_0 \cap \mathcal{F}_1$, 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: | ||
Revision as of 20:15, 4 June 2011
Shishikura writes[1] in
The statements of Shishikura
Shishikura writes[1] 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 $f_0\in \mathcal{N}_0 \cap \mathcal{F}_1$, 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 $f\to f_0$ $$\ph_f\to\ph_0\quad\text{and}\quad \tilde{\mathcal{R}}_f + \frac{1}{\alpha(f)} \to \tilde{\mathcal{E}}_{f_0}$$