Perturbed Fatou coordinates - Revision history

Bo198214: moved to singular

2011-06-05T14:42:08Z

moved to singular

Bo198214: Added some more explanation

2011-06-05T11:34:06Z

Added some more explanation

Bo198214: Moved out Shishikura

2011-06-04T20:27:51Z

Moved out Shishikura

Bo198214: corrected typos

2011-06-04T20:20:48Z

corrected typos

Bo198214: Added definition of F_0 and F_1

2011-06-04T20:15:27Z

Added definition of F_0 and F_1

Bo198214: This is just the plain theorem, we need to prepend all contained definitions

2011-06-04T19:54:16Z

This is just the plain theorem, we need to prepend all contained definitions

New page

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:

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 $f\to f_0$ $$\ph_f\to\ph_0\quad\text{and}\quad \tilde{\mathcal{R}}_f + \frac{1}{\alpha(f)} \to \tilde{\mathcal{E}}_{f_0}$$

<references/>

← Older revision		Revision as of 14:42, 5 June 2011
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−	~~While~~ [[~~Fatou coordinates]] are the (injective parts of the) [[principal Abel function]]s of a holomorphic function $f$ at a given fixpoint,~~	+	#REDIRECT [[perturbed Fatou coordinate]]
−	~~for simplicity this fixpoint is assumed to be at 0:~~
−	~~$$f(z)= z + c_{m+1} z^{m+1} + o(z^{m+1}),$$~~
−
−	'perturbed' Fatou ~~coordinates refer to a function that is 'disturbed' by some small $c_0\in\C$:~~
−	~~$$f(z)=c_0 + z + c_{m+1} z^{m+1} + o(z^{m+1})$$~~
−
−	~~This function usually has $2m$ fixpoints in a vicinity of 0.~~
−
−	~~For the details of the case $m=1$ I refer to [[Shishikura_perturbed_Fatou_coordinates\|Shishikuras presentation~~]].

← Older revision		Revision as of 11:34, 5 June 2011
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−	~~I will use~~ [[~~Shishikura_perturbed_Fatou_coordinates\|this~~ (~~emerging~~) ~~page~~]] to ~~quote Shishikura~~, ~~and use perhaps this article~~ to ~~draw~~ some ~~conclusions later~~.	+	While [[Fatou coordinates]] are the (injective parts of the) [[principal Abel function]]s of a holomorphic function $f$ at a given fixpoint,
		+	for simplicity this fixpoint is assumed to be at 0:
		+	$$f(z)= z + c_{m+1} z^{m+1} + o(z^{m+1}),$$
		+
		+	'perturbed' Fatou coordinates refer to a function that is 'disturbed' by some small $c_0\in\C$:
		+	$$f(z)=c_0 + z + c_{m+1} z^{m+1} + o(z^{m+1})$$
		+
		+	This function usually has $2m$ fixpoints in a vicinity of 0.
		+
		+	For the details of the case $m=1$ I refer to [[Shishikura_perturbed_Fatou_coordinates\|Shishikuras presentation]].

← Older revision		Revision as of 20:27, 4 June 2011
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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/>~~

@@ Line 5: / Line 5: @@
 === 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
 $$ \mathcal{F}_0 = \{ f_0 \colon f_0 \quad\text{is defined and analytic in a neighborhood of 0}\quad f_0(0)=0, f_0'(0)=1, f_0''(0)=1\}$$
@@ Line 15: / Line 15: @@
 $$ \mathcal{F}_1 = \{ f\in \mathcal{F}\colon |\arg(\alpha(f))| < \frac{\pi}{4}\} $$
-=== proposition 4.4.1 p. 356 ===
+=== 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 $f\in \mathcal{N}_0 \cap \mathcal{F}_1$, 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 $\Im(w)\to -\infty$.

← Older revision		Revision as of 20:15, 4 June 2011
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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
		+
		+	== 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 $f_0\in \mathcal{N}_0 \cap \mathcal{F}_1$, 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: