Difference between revisions of "Shishikura 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> the following | 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 | + | == $\mathcal{F}_0$ == |
| + | (p. 327) | ||
| + | |||
If $f_0''(0)\neq 0$ by another coordinate change we may assume that $f_0''(0)=1$, so define | If $f_0''(0)\neq 0$ by another coordinate change we may assume that $f_0''(0)=1$, so define | ||
$$ \mathcal{F}_0 = \{ f_0 \colon f_0 \quad\text{is defined and analytic in a neighborhood of 0}\quad f_0(0)=0, f_0'(0)=1, f_0''(0)=1\}$$ | $$ \mathcal{F}_0 = \{ f_0 \colon f_0 \quad\text{is defined and analytic in a neighborhood of 0}\quad f_0(0)=0, f_0'(0)=1, f_0''(0)=1\}$$ | ||
| − | == | + | == Neighborhood of $f$ in the compact-open topology with domain of definition == |
In what follows a map $f$ is always associated with its domain of definition $D(f)$. That is two maps are considered as distinct if they have distinct domains, even if the one is an extension of the other. A neighborhood of an analytic map $f\colon D(f)\to\overline{\C}$ is a set containing | In what follows a map $f$ is always associated with its domain of definition $D(f)$. That is two maps are considered as distinct if they have distinct domains, even if the one is an extension of the other. A neighborhood of an analytic map $f\colon D(f)\to\overline{\C}$ is a set containing | ||
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A sequence $(f_n)$ converges to $f$ in this topology if and only if for any compact set $K\subset D(f)$ there exists $n_0$ such that $K\subset D(f_n)$ for all $n\ge n_0$ and $f_n|_K\to f|_K$ uniformly on $K$ as $n\to\infty$. The system of these neighborhoods define the "compact-open topology together with the domain of defintion", which is unfortunately not Hausdorff, since an extension of $f$ is contained in any neighborhood of $f$. | A sequence $(f_n)$ converges to $f$ in this topology if and only if for any compact set $K\subset D(f)$ there exists $n_0$ such that $K\subset D(f_n)$ for all $n\ge n_0$ and $f_n|_K\to f|_K$ uniformly on $K$ as $n\to\infty$. The system of these neighborhoods define the "compact-open topology together with the domain of defintion", which is unfortunately not Hausdorff, since an extension of $f$ is contained in any neighborhood of $f$. | ||
| − | === | + | For $b_1,b_2\in\C$ with $\Re(b_1)<\Re(b_2)$ define |
| + | $$Q(b_1,b_2)=\{z\in\C\colon \Re(z-b_1)>-|\Im(z-b_1)|, \Re(z-b_2)>|\Im(z-b_2)|\}$$ | ||
| + | If $b_1=-\infty$ (resp. $b_2=\infty$) then the corresponding condition should be removed. | ||
| + | |||
| + | == Proposition 2.5.2 == | ||
| + | |||
| + | Let $f$ be a holomorphic function defined on $Q=Q(b_1,b_2)$ with $\Re(b_2)>\Re(b_1)+2$ (here $b_1$ or $b_2$ may be $-\infty$ or $\infty$). Suppose | ||
| + | $$|F(z)-(z+1)|<\frac{1}{4},\quad |F'(z)-(z+1)|<\frac{1}{4}\quad \text{for all}\quad z\in Q$$ | ||
| + | |||
| + | == $\mathcal{F}$ == | ||
| + | (p. 339) | ||
| + | |||
$$ \mathcal{F} = \{ f \colon f \quad\text{is defined and analytic in a neighborhood of 0}\quad f(0)=0, f_0'(0)\neq 0\}$$ | $$ \mathcal{F} = \{ f \colon f \quad\text{is defined and analytic in a neighborhood of 0}\quad f(0)=0, f_0'(0)\neq 0\}$$ | ||
For $f\in \mathcal{F}$ we express the derivative | For $f\in \mathcal{F}$ we express the derivative | ||
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$$ \mathcal{F}_1 = \{ f\in \mathcal{F}\colon |\arg(\alpha(f))| < \frac{\pi}{4}\} $$ | $$ \mathcal{F}_1 = \{ f\in \mathcal{F}\colon |\arg(\alpha(f))| < \frac{\pi}{4}\} $$ | ||
| − | === p. 340 | + | |
| + | == $\sigma(f)$ == | ||
| + | (p. 340) | ||
Let $\sigma(f)$ be the other fixpoint of $f$ (besides 0). | Let $\sigma(f)$ be the other fixpoint of $f$ (besides 0). | ||
| − | + | == Proposition 4.4.1 == | |
| + | (p. 356) | ||
Let $f_0\in\mathcal{F}_0$, there exists a neighborhood $\mathcal{N}_0$ of $f_0$ (in the previously described "compact-open topology with domain of definition"), 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: | Let $f_0\in\mathcal{F}_0$, there exists a neighborhood $\mathcal{N}_0$ of $f_0$ (in the previously described "compact-open topology with domain of definition"), 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: | ||
Revision as of 05:54, 5 June 2011
Shishikura writes[1] the following
$\mathcal{F}_0$
(p. 327)
If $f_0''(0)\neq 0$ by another coordinate change we may assume that $f_0''(0)=1$, so define $$ \mathcal{F}_0 = \{ f_0 \colon f_0 \quad\text{is defined and analytic in a neighborhood of 0}\quad f_0(0)=0, f_0'(0)=1, f_0''(0)=1\}$$
Neighborhood of $f$ in the compact-open topology with domain of definition
In what follows a map $f$ is always associated with its domain of definition $D(f)$. That is two maps are considered as distinct if they have distinct domains, even if the one is an extension of the other. A neighborhood of an analytic map $f\colon D(f)\to\overline{\C}$ is a set containing $$\{ g\colon D(g)\to\overline{\C}| g \quad\text{is analytic}\quad, D(g)\supset K, \sup_{z\in K} d(f(z)-g(z))<\eps \} $$ where $K$ is a compact set in $D(f)$, $\eps>0$ and $d(.,.)$ is the spherical metric. A sequence $(f_n)$ converges to $f$ in this topology if and only if for any compact set $K\subset D(f)$ there exists $n_0$ such that $K\subset D(f_n)$ for all $n\ge n_0$ and $f_n|_K\to f|_K$ uniformly on $K$ as $n\to\infty$. The system of these neighborhoods define the "compact-open topology together with the domain of defintion", which is unfortunately not Hausdorff, since an extension of $f$ is contained in any neighborhood of $f$.
For $b_1,b_2\in\C$ with $\Re(b_1)<\Re(b_2)$ define $$Q(b_1,b_2)=\{z\in\C\colon \Re(z-b_1)>-|\Im(z-b_1)|, \Re(z-b_2)>|\Im(z-b_2)|\}$$ If $b_1=-\infty$ (resp. $b_2=\infty$) then the corresponding condition should be removed.
Proposition 2.5.2
Let $f$ be a holomorphic function defined on $Q=Q(b_1,b_2)$ with $\Re(b_2)>\Re(b_1)+2$ (here $b_1$ or $b_2$ may be $-\infty$ or $\infty$). Suppose $$|F(z)-(z+1)|<\frac{1}{4},\quad |F'(z)-(z+1)|<\frac{1}{4}\quad \text{for all}\quad z\in Q$$
$\mathcal{F}$
(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 $$ \mathcal{F}_1 = \{ f\in \mathcal{F}\colon |\arg(\alpha(f))| < \frac{\pi}{4}\} $$
$\sigma(f)$
(p. 340)
Let $\sigma(f)$ be the other fixpoint of $f$ (besides 0).
Proposition 4.4.1
(p. 356)
Let $f_0\in\mathcal{F}_0$, there exists a neighborhood $\mathcal{N}_0$ of $f_0$ (in the previously described "compact-open topology with domain of definition"), 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 $f\to f_0$ $$\ph_f\to\ph_0\quad\text{and}\quad \tilde{\mathcal{R}}_f + \frac{1}{\alpha(f)} \to \tilde{\mathcal{E}}_{f_0}$$
- ↑ 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)