Shishikura perturbed Fatou coordinates

Shishikura writes 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

[ p. 332 ]

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 $$\{ g \colon g \quad\text{is analytic},\quad D(g)\supset K, \sup_{z\in K} d(f(z)-g(z))<\eps \} $$ 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

[p. 333]

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 $$|F(z)-(z+1)|<\frac{1}{4},\quad |F'(z)-(z+1)|<\frac{1}{4}\quad \text{for all}\quad z\in Q.$$

Then

1. $F$ is univalent [means injective] on $Q$.
2. Let $z_0\in Q$ be a point such that $\Re(b_1)<\Re(z_0)<\Re(b_2)-\frac{5}{4}$. Denote by $S$ the closed region (a strip) bounded by the two curves $\ell=\{z_0+i y\colon y\in\R\}$ and $F(\ell)$. Then for any $z\in Q$ there exist a unique $n\in \Z$ such that $F^n(z)$ is defined and belongs to $S-F(\ell)$.
3. There exists an univalent function $\Phi\colon Q\to \C$ satisfying $$\Phi(F(z))=\Phi(z)+1$$ whenever both sides are defined. Moreover $\Phi$ is unique up to an addition of a constant.
4. Fix a point $z_0\in Q$. If we normalize $\Phi$ by $\Phi(z_0)=0$ then the correspondence $F\mapsto \Phi$ is continuous with respect to the compact-open topology.

$\mathcal{F}$, $\mathcal{F}_1$ and $\alpha(f)$

[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 $$ \mathcal{F}_1 = \{ f\in \mathcal{F}\colon |\arg(\alpha(f))| < \frac{\pi}{4}\} $$

Comments

By the definition we have $$ \begin{align*} \log(f'(0))&=2\pi i\alpha(f)=2\pi i re^{i\beta}=2\pi r e^{i(\beta+\frac{\pi}{2})}\\ \arg(\log(f'(0))&=\beta+\frac{\pi}{2} \end{align*}$$ so the condition for $\mathcal{F}_1$ is $|\arg(\log(f'(0)))-\frac{\pi}{2}|<\frac{\pi}{4}$ or $$\frac{\pi}{4} < \arg(\log(f'(0)) < \frac{3\pi}{4}$$

$\sigma(f)$

[p. 340]

Let $\sigma(f)$ be the other fixpoint of $f$ (besides 0).

...

$$\sigma(f) = 2\pi i \alpha(f)(1+o(1)), \quad\text{as}\quad f\to f_0$$

Comments

By the comment before, we can say that $\sigma(f)$ lies roughly in the sector of angle $(\frac{\pi}{4},\frac{3\pi}{4})$ above the tip 0.

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:

1. $S_{\pm,f}$ 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 $S_{-,f}\cap S_{+,f} = \{0,\sigma(f)\}$;
2. $\Phi_{\pm,f}$ is defined, analytic and injective in a neighborhood of $S'_{\pm,f} = S_{\pm,f}\setminus \{0,\sigma(f)\}$;
3. 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 $n$, \[ \Phi_{+,f}(f^n(z)) = \Phi_{-,f}(z) - \frac{1}{\alpha(f)} + n \]
4. 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

[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:

1. $D(\ph_f)$ 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 $\Im(w)\to -\infty$.
2. $D(\tilde{\mathcal{R}}_f)$ 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 $\Im(w)\to -\infty$.
3. 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 $n>0$.
4. When $f\in\mathcal{F_1}$ and $f\to f_0$ $$\ph_f\to\ph_0\quad\text{and}\quad \tilde{\mathcal{R}}_f + \frac{1}{\alpha(f)} \to \tilde{\mathcal{E}}_{f_0}$$

Proposition A.1

[p. 360]

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 $s\in\Delta$.