Shishikura perturbed Fatou coordinates

Shishikura writes 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. 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 $$\{ g\colon D(g)\to\overline{\C}| g \quad\text{is analytic}\quad, D(g)\supset K, \sup_{z\in K} d(f(z)-g(z))<\eps \} $$ 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 $f$.

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