Shishikura perturbed Fatou coordinates - Revision history

Bo198214 at 08:42, 16 July 2019

2019-07-16T08:42:42Z

Bo198214: /* Proposition 3.2.2 */ used wiki numbering

2013-02-19T21:52:44Z

Proposition 3.2.2: used wiki numbering

Bo198214: /* Proposition A.1 */ added page

2013-02-19T21:50:33Z

Proposition A.1: added page

Bo198214: /* Proposition A.1 */ fixed typo

2013-02-19T21:49:26Z

Proposition A.1: fixed typo

Bo198214: /* Proposition A.1 */ removed unnecessary notation

2013-02-19T21:48:47Z

Proposition A.1: removed unnecessary notation

Bo198214: /* Proposition A.1 */ wrote Proposition A.1

2013-02-19T21:46:58Z

Proposition A.1: wrote Proposition A.1

Bo198214: added proposition A.1

2013-02-19T21:40:18Z

added proposition A.1

Bo198214: /* Proposition 3.2.2 */

2013-02-19T21:39:06Z

Proposition 3.2.2

Bo198214: /* Proposition 3.2.2 */ added page number

2013-02-19T21:38:40Z

Proposition 3.2.2: added page number

Bo198214: /* Proposition 3.2.2 */ fixed some TeX

2013-02-19T21:35:34Z

Proposition 3.2.2: fixed some TeX

@@ Line 69: / Line 69: @@
 Let $f_0\in\mathcal{F}_0$. There exists a neighborhood $\mathcal{N}_0$ of $f_0$ (in the sense of the topology defined in 2.5.1) such that if $f\in\mathcal{N}_0\cap \mathcal{F}_1$, then there exist closed Jordan domains $S_{-,f}$, $S_{+,f}$ and analytic functions $\Phi_{-,f}$, $\Phi_{+,f}$ satisfying the following:
-# $S_{+,f}$ is bounded by an arc $\ell_{\pm}$ and its image $f(\ell_{\pm})$ such that $\ell_{\pm}$ joins two fixed points $\{0,\sigma(f)\}$, $\ell_{\pm}\cap f(\ell_{\pm}) = \{0,\sigma(f)\}$ and $S_{-,f}\cap S_{+,f} = \{0,\sigma(f)\}$;
+# $S_{\pm,f}$ is bounded by an arc $\ell_{\pm}$ and its image $f(\ell_{\pm})$ such that $\ell_{\pm}$ joins two fixed points $\{0,\sigma(f)\}$, $\ell_{\pm}\cap f(\ell_{\pm}) = \{0,\sigma(f)\}$ and $S_{-,f}\cap S_{+,f} = \{0,\sigma(f)\}$;
 # $\Phi_{\pm,f}$ is defined, analytic and injective in a neighborhood of $S'_{\pm,f} = S_{\pm,f}\setminus \{0,\sigma(f)\}$;
 # if $z\in S'_{-,f}$ then there is an $n\ge 1$ such that $f^n(z)\in S'_{+,f}$ and for the smallest such $n$, \[ \Phi_{+,f}(f^n(z)) = \Phi_{-,f}(z) - \frac{1}{\alpha(f)} + n \]

@@ Line 69: / Line 69: @@
 Let $f_0\in\mathcal{F}_0$. There exists a neighborhood $\mathcal{N}_0$ of $f_0$ (in the sense of the topology defined in 2.5.1) such that if $f\in\mathcal{N}_0\cap \mathcal{F}_1$, then there exist closed Jordan domains $S_{-,f}$, $S_{+,f}$ and analytic functions $\Phi_{-,f}$, $\Phi_{+,f}$ satisfying the following:
-(i) $S_{+,f}$ is bounded by an arc $\ell_{\pm}$ and its image $f(\ell_{\pm})$ such that $\ell_{\pm}$ joins two fixed points $\{0,\sigma(f)\}$, $\ell_{\pm}\cap f(\ell_{\pm}) = \{0,\sigma(f)\}$ and $S_{-,f}\cap S_{+,f} = \{0,\sigma(f)\}$;
+# $S_{+,f}$ is bounded by an arc $\ell_{\pm}$ and its image $f(\ell_{\pm})$ such that $\ell_{\pm}$ joins two fixed points $\{0,\sigma(f)\}$, $\ell_{\pm}\cap f(\ell_{\pm}) = \{0,\sigma(f)\}$ and $S_{-,f}\cap S_{+,f} = \{0,\sigma(f)\}$;
+# $\Phi_{\pm,f}$ is defined, analytic and injective in a neighborhood of $S'_{\pm,f} = S_{\pm,f}\setminus \{0,\sigma(f)\}$;
-(ii) $\Phi_{\pm,f}$ is defined, analytic and injective in a neighborhood of $S'_{\pm,f} = S_{\pm,f}\setminus \{0,\sigma(f)\}$;
+# if $z\in S'_{-,f}$ then there is an $n\ge 1$ such that $f^n(z)\in S'_{+,f}$ and for the smallest such $n$, \[ \Phi_{+,f}(f^n(z)) = \Phi_{-,f}(z) - \frac{1}{\alpha(f)} + n \]
+# when $f\in\mathcal{F}_1$ tends to $f_0$, $S_{\pm,f}\to S_{\pm,0}\cup \{0\}$ in the Hausdorff metric, and $\Phi_{\pm,f}\to \Phi_{\pm,0}$ in the topology defined in 2.5.1.
 == Proposition 4.4.1 ==

@@ Line 88: / Line 88: @@
 == Proposition A.1 ==
 In the settings of Proposition 3.2.2 (resp. 2.5.2), assume in addtion that $f_s(z)$ (resp. $F_s(z)$) is a family of holomorphic maps for $s\in \Delta \cup \{0\} \subset \mathbb{C}^k$, with $\Delta\subset\mathbb{C}^k$ an open connected set containing 0 in the boundary, such that $f_s$ (resp. $F_s(z)$) is continuous for $s\in\Delta\cup \{0\}$ and analytic in $s\in\Delta$, then the Fatou coordinates $\Phi_{\pm,f_s}$ (resp. $\Phi_{F_s}$) depend analytically on $s$ for $s\in\Delta$.
 <references/>

@@ Line 88: / Line 88: @@
 == Proposition A.1 ==
-In the settings of Proposition 3.2.2 (resp. 2.5.2), assume in addtion that $f_s(z)$ (resp. $F_s(z)$) is a family of holomorphic maps for $s\in \Delta \cup \{0\} \subset \mathbb{C}^k$, with $\Delta\subset\mathbb{C}^k$ an open connected set containing 0 in the boundary, such that $f_s$ (resp. $F_s(z)$) is continuous for $s\in\Delta\cup \{0\}$ and analytic in $s\in\Delta$, the the Fatou coordinates $\Phi_{\pm,f_s}$ (resp. $\Phi_{F_s}$) depend analytically on $s$ for $s\in\Delta$.
+In the settings of Proposition 3.2.2 (resp. 2.5.2), assume in addtion that $f_s(z)$ (resp. $F_s(z)$) is a family of holomorphic maps for $s\in \Delta \cup \{0\} \subset \mathbb{C}^k$, with $\Delta\subset\mathbb{C}^k$ an open connected set containing 0 in the boundary, such that $f_s$ (resp. $F_s(z)$) is continuous for $s\in\Delta\cup \{0\}$ and analytic in $s\in\Delta$, then the Fatou coordinates $\Phi_{\pm,f_s}$ (resp. $\Phi_{F_s}$) depend analytically on $s$ for $s\in\Delta$.
 <references/>

@@ Line 88: / Line 88: @@
 == Proposition A.1 ==
-In the settings of Proposition 3.2.2 (resp. 2.5.2), assume in addtion that $f_s(z)$ (resp. $F_s(z)$) is a family of holomorphic maps for $s\in \Delta \cup \{0\} \subset \mathbb{C}^k$, with $\Delta\subset\mathbb{C}^k$ an open connected set containing 0 in the boundary, such that $f_s$ (resp. $F_s(z)$) is continuous for $s\in\Delta\cup \{0\}$ and analytic in $s\in\Delta$, the the Fatou coordinates $\Phi_{\pm,f_s} := \Phi_{\pm,s}$ (resp. $\Phi_s := \Phi_{F_s}$) depend analytically on $s$ for $s\in\Delta$.
+In the settings of Proposition 3.2.2 (resp. 2.5.2), assume in addtion that $f_s(z)$ (resp. $F_s(z)$) is a family of holomorphic maps for $s\in \Delta \cup \{0\} \subset \mathbb{C}^k$, with $\Delta\subset\mathbb{C}^k$ an open connected set containing 0 in the boundary, such that $f_s$ (resp. $F_s(z)$) is continuous for $s\in\Delta\cup \{0\}$ and analytic in $s\in\Delta$, the the Fatou coordinates $\Phi_{\pm,f_s}$ (resp. $\Phi_{F_s}$) depend analytically on $s$ for $s\in\Delta$.
 <references/>

← Older revision		Revision as of 21:46, 19 February 2013
Line 88:		Line 88:

	== Proposition A.1 ==		== Proposition A.1 ==
		+	In the settings of Proposition 3.2.2 (resp. 2.5.2), assume in addtion that $f_s(z)$ (resp. $F_s(z)$) is a family of holomorphic maps for $s\in \Delta \cup \{0\} \subset \mathbb{C}^k$, with $\Delta\subset\mathbb{C}^k$ an open connected set containing 0 in the boundary, such that $f_s$ (resp. $F_s(z)$) is continuous for $s\in\Delta\cup \{0\}$ and analytic in $s\in\Delta$, the the Fatou coordinates $\Phi_{\pm,f_s} := \Phi_{\pm,s}$ (resp. $\Phi_s := \Phi_{F_s}$) depend analytically on $s$ for $s\in\Delta$.

	<references/>		<references/>

@@ Line 86: / Line 86: @@
 # 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 66: / Line 66: @@
 == Proposition 3.2.2 ==
 [p. 340]
 Let $f_0\in\mathcal{F}_0$. There exists a neighborhood $\mathcal{N}_0$ of $f_0$ (in the sense of the topology defined in 2.5.1) such that if $f\in\mathcal{N}_0\cap \mathcal{F}_1$, then there exist closed Jordan domains $S_{-,f}$, $S_{+,f}$ and analytic functions $\Phi_{-,f}$, $\Phi_{+,f}$ satisfying the following:

← Older revision		Revision as of 21:38, 19 February 2013
Line 65:		Line 65:

	== Proposition 3.2.2 ==		== Proposition 3.2.2 ==
		+	[p. 340]
	Let $f_0\in\mathcal{F}_0$. There exists a neighborhood $\mathcal{N}_0$ of $f_0$ (in the sense of the topology defined in 2.5.1) such that if $f\in\mathcal{N}_0\cap \mathcal{F}_1$, then there exist closed Jordan domains $S_{-,f}$, $S_{+,f}$ and analytic functions $\Phi_{-,f}$, $\Phi_{+,f}$ satisfying the following:		Let $f_0\in\mathcal{F}_0$. There exists a neighborhood $\mathcal{N}_0$ of $f_0$ (in the sense of the topology defined in 2.5.1) such that if $f\in\mathcal{N}_0\cap \mathcal{F}_1$, then there exist closed Jordan domains $S_{-,f}$, $S_{+,f}$ and analytic functions $\Phi_{-,f}$, $\Phi_{+,f}$ satisfying the following: