# Non-degenerate near-parabolic renormalization

Alex Kapiamba

###### Abstract.

Invariant classes under parabolic and near-parabolic renormalization have
proved extremely useful for studying the dynamics of polynomials. The first
such class was introduced by Inou-Shishikura to study quadratic polynomials;
their argument has been extended to the unicritical cubic case by Yang.
Recently, Chéritat has constructed new invariant classes for these
renormalizations, general enough to be applied to unicritical polynomials of
arbitrary degree. Usually the parabolic and near-parabolic renormalizations
are only considered for maps which have a fixed point with multiplier close to
$1$, though it is well-known that similar phenomena occur when the multiplier
is close to any root of unity. In this paper we define the parabolic and near-
parabolic renormalization operators in the general setting, and use Chéritat’s
argument to produce invariant classes for these operators.

## Introduction

Let $f(z)$ be a holomorphic function defined on a neighborhood of
$0\in\mathbb{C}$ which fixes $z=0$. The fixed point $0$ is called parabolic if
the multiplier $\lambda=f^{\prime}(0)$ is a root of unity. For now we assume
$\lambda=1$, in this case we say the fixed point is $1$-parabolic. In addition
we assume that $f^{\prime\prime}(0)\neq 0$, in this case we say that the
parabolic fixed point is non-degenerate.

The local dynamics of $f$ near $z=0$ are relatively simple, orbits are
attracted towards $0$ on one side and repelled away from $0$ on another.
However, for a holomorphic function $h$ which is a perturbation of $f$, the
dynamics can be drastically more complicated. Orbits are still attracted
towards $0$ on one side and repelled away from $0$ on another, but in this
perturbed setting the orbits of points on the attracting side may transfer to
the repelling side. This phenomenon, often called parabolic implosion, was
first studied by Douady-Hubbard [DH84] [DH85] and Lavaurs [Lav89] and has
found many applications in holomorphic dynamics, see for example [Dou94],
[Shi98], [Shi00], [Ast+16], [Kap21].

In proving that the boundary of the Mandelbrot set has Hausdorff dimension $2$
[Shi98], Shishikura introduced a pair of renormalization operators which
provide another lens to study parabolic implosion: the parabolic and near-
parabolic renormalizations. These operators are defined by quotienting the
dynamics $f$ and $h$ respectively near $z=0$. In some cases, the parabolic or
near-parabolic renormalization of $f$ or $h$ respectively can either have a
parabolic fixed point or be a perturbation of a map with a parabolic fixed
point, allowing for the construction of successive parabolic or near-parabolic
renormalizations. Shishikura introduced a class of maps which is invariant
under parabolic renormalization [Shi98], and Inou-Shishikura introduced a
class of maps, called the Inou-Shishikura class, which is invariant under
near-parabolic renormalization [IS08]. The latter result has had several
remarkable applications: it is used by Buff-Chéritat to prove the existence of
quadratic Julia sets of positive measure [BC12], by Cheraghi-Chéritat to
partially resolve the Marmi-Moussa-Yoccoz conjecture [CC15], and by Cheraghi-
Shishikura to make progress towards the celebrated MLC conjecture [CS15].
There are numerous other applications of the Inou-Shishikura result, see for
example [AC18], [Che13], [Che17], [Che19], [SY16].

All maps in the the Inou-Shishikura class are of the form $e^{2\pi
i\alpha}\cdot G\circ\varphi^{-1}$, where $\alpha$ is a complex number close to
$0$, $G$ is some fixed analytic function, and $\varphi$ is a normalized
univalent function defined on some fixed domain $V$. Maps in the Inou-
Shishikura class all have a critical point of local degree $2$, so most
applications of the class are found in quadratic polynomials. Yang has
modified the argument of Inou-Shishikura to produce a class of maps, all with
a critical point of local degree $3$, which is invariant under parabolic and
near-parabolic renormalization [Yan15], allowing for applications to
unicritical cubic polynomials. Chéritat has introduced classes of maps which
are also invariant under near-parabolic renormalization [Ché22], all
corresponding to different choices of the domain function $G$ and domain $V$.
The added benefit Chéritat’s construction is that the argument is flexible
enough to build classes which can be applied to unicritical polynomials of
arbitrary degree.

While it is well-known that similar parabolic implosion phenomenon occur if we
instead allow $\lambda$ to be an arbitrary root of unity, the bulk of the
theory in the literature is devoted to the the non-degnerate $1$-parabolic
case. When $\lambda$ is a primitive $q$-th root of unity the local dynamics
remain relatively simple: orbits are attracted towards $0$ along $\nu$
distinct directions and repelled away from $0$ along $\nu$ other distinct
directions where $\nu$ is an integer multiple of $q$. The parabolic fixed
point is said to be non-degenerate if $\nu=q$. While qualitatively the
implosion that occurs for a perturbation $h$ of $f$ is analogous to the
$\lambda=1$ case, there are a few subtitles which arise under careful
analysis. Oudkerk studied the parabolic implosion of degenerate $1$-parabolic
fixed points [Oud02], which encompasses the theory of general non-degenerate
parabolic implosion as $f^{q}$ has a degenerate $1$-parabolic fixed point.
However, in the degenerate setting studied by Oudkerk the bifurcations which
occur for perturbations of $f$ can become quite complicated; in the symmetry
induced on $f^{q}$ by the first iterate of $f$ allows for a simpler
description. The first goal of this paper is to provide a reference for the
theory of parabolic implosion in the non-degenerate setting which avoids the
added complications in [Oud02] that arise from degenerate parabolic fixed
points.

In the setting of non-degenerate parabolic implosion, we can define the
parabolic and near-parabolic renormalizations in analogy to the definition in
the $1$-parabolic case. It is natural to ask, what are invariant classes of
these renormalizations? The argument in [IS08] to prove the invariance of the
Inou-Shishikura class under near-parabolic renormalization relies on precise
and delicate computations, which makes it difficult to generalize immediately
to the non-degenerate setting. The second goal of this paper is to observe
that the argument in [Ché22] can be applied to the general non-degenerate
case. In particular, it follows that:

######  Theorem (Imprecise version).

For any root of unity, there are classes of maps which are invariant under the
corresponding parabolic or near-parabolic renormalization.

A precise version of the above theorem is given in theorem 3.4 and corollary
3.7.

As the traditional parabolic and near-parabolic renormalizations are defined
only in the non-degenerate $1$-parabolic setting, many of the applications of
the Inou-Shishikura class are restricted to maps which have an indifferent
fixed point of high-type, i.e. the multiplier of the fixed point is $e^{2\pi
i\theta}$ and all the entries of the continued fraction expansion of $\theta$
are large. See for example [CC15], [CS15], [SY16], [Che17]. The invariant
classes constructed in this paper allow us to use near-parabolic
renormalization to instead study maps which have an indifferent fixed point of
what we call valley-type, where the continued fraction expansion of $\theta$
has large entries a bounded distance away from each other. We can expect that
the applications of the Inou-Shishikura class to the high-type setting can be
generalized to the valley-type setting.

This paper is organized as follows. In $\S 1$ we present some elementary facts
about continued fractions which will be important to our analysis. In $\S 2$
we describe parabolic implosion in the general non-degenerate setting and
define the associated parabolic and near-parabolic renormalizations. In $\S 3$
we construct the corresponding invariant classes.

Acknowledgments. This research was supported in part by the NSF and the Ford
Foundation.

##  1\. Modified continued fractions

For $n\in\mathbb{Z}_{\geq 0}\cup\\{\infty\\}$, we define a modified continued
fraction of height $n$ to be a sequence

$\omega=\langle
a_{j}:\varepsilon_{j}\rangle_{j=1}^{n}=\langle(a_{1}:\varepsilon_{1}),(a_{2}:\varepsilon_{2}),\dots,(a_{n}:\varepsilon_{n})\rangle,$

where each $a_{j}$ is an integer larger than one and
$\varepsilon_{j}\in\\{-1,+1\\}$. Note that the empty sequence, which we denote
$\emptyset$, is the unique modified continued fraction of height $0$. For all
$n$, let $\mathcal{Q}_{n}$ be the set of all modified continued fractions of
height $n$, and set

$\mathcal{Q}:=\bigcup_{n\in\mathbb{Z}_{\geq 0}}\mathcal{Q}_{n}.$

For all $n$ and $\omega=\langle
a_{j}:\varepsilon_{j}\rangle_{j=1}^{n}\in\mathcal{Q}_{n}$ we define the
integers

$\displaystyle\mathfrak{p}_{-1}(\omega)=1,$
$\displaystyle\mathfrak{p}_{0}(0)=0,\hskip 14.22636pt$
$\displaystyle\mathfrak{p}_{j}(\omega)=a_{j}\mathfrak{p}_{j-1}(\omega)+\varepsilon_{j}\mathfrak{p}_{j-2}(\omega)$
$\displaystyle\text{ for }1\leq j\leq n,$
$\displaystyle\mathfrak{q}_{-1}(\omega)=0,$
$\displaystyle\mathfrak{q}_{0}(0)=1,\hskip 14.22636pt$
$\displaystyle\mathfrak{q}_{j}(\omega)=a_{j}\mathfrak{q}_{j-1}(\omega)+\varepsilon_{j}\mathfrak{q}_{j-2}(\omega)$
$\displaystyle\text{ for }1\leq j\leq n.$

When $n$ is finite, we set $\mathfrak{p}(\omega)=\mathfrak{p}_{n}(\omega)$,
$\mathfrak{q}(\omega)=\mathfrak{q}_{n}(\omega)$. We also define the signature
of $\omega$ to be

$\mathcal{E}(\omega)=(-1)^{n}\prod_{j=1}^{n}\varepsilon_{j},$

using the convention that the empty product is equal to one.

######  Proposition 1.1.

For any $n\in\mathbb{Z}_{\geq 0}$ and $\omega\in\mathcal{Q}_{n}$,

$\mathcal{E}(\omega)=\mathfrak{p}_{n-1}(\omega)\mathfrak{q}_{n}(\omega)-\mathfrak{p}_{n}(\omega)\mathfrak{q}_{n-1}(\omega)$

###### Proof.

This follows from a straightforward induction on $n\geq 0$. ∎

For $n\in\mathbb{Z}_{\geq 0}$ and $\omega\in\mathcal{Q}_{n}$, we define the
Möbius transformation $\mu_{\omega}:\hat{\mathbb{C}}\to\hat{\mathbb{C}}$ by

(1) $\mu_{\omega}(z):=\frac{\mathfrak{p}_{n}(\omega)+\mathcal{E}(\omega)\cdot
z\mathfrak{p}_{n-1}(\omega)}{\mathfrak{q}_{n}(\omega)+\mathcal{E}(\omega)\cdot
z\mathfrak{q}_{n-1}(\omega)}=\cfrac{\varepsilon_{1}}{a_{1}+\cfrac{\varepsilon_{2}}{\ddots+\cfrac{\varepsilon_{n}}{a_{n}+\mathcal{E}(\omega)\cdot
z}}}.$

For $\omega\in\mathcal{Q}_{\infty}$ we will similarly define
$\mu_{\omega}:\hat{\mathbb{C}}\to\hat{\mathbb{C}}$ to be the constant function

$\mu_{\omega}(z):=\cfrac{\varepsilon_{1}}{a_{1}+\cfrac{\varepsilon_{2}}{\ddots}}.$

We will also set the signature of every infinite height
$\omega\in\mathcal{Q}_{\infty}$ to be $\mathcal{E}(\omega):=0$. Note that the
function $\omega\mapsto\mu_{\omega}(0)$ maps $\mathcal{Q}$ onto the rational
numbers in $(-1,1)$, and maps $\mathcal{Q}_{\infty}$ onto the irrational
numbers in $(-1,1)$.

We define the conjugate of $\omega\in{\mathcal{Q}}_{n}$ to be the extended
modified continued fraction $\omega^{*}=\langle
a_{j}^{\prime}:\varepsilon_{j}^{\prime}\rangle_{j=1}^{n}$, where
$a_{j}^{\prime}=a_{j}$ for all $1\leq j\leq n$,
$\varepsilon_{1}^{\prime}=-\varepsilon_{1}$, and
$\varepsilon_{j}^{\prime}=\varepsilon_{j}$ for all $1<j\leq n.$

######  Proposition 1.2.

For all $\omega\in\mathcal{Q}$, $\mu_{\omega^{*}}(z)=-\mu_{\omega}(-z).$

###### Proof.

It follows from the definition that $\emptyset^{*}=\emptyset$, so
$\mu_{\emptyset^{*}}(z)=z=-\mu_{\emptyset}(-z)$ automatically. If
$\omega\in\mathcal{Q}_{n}$ for some $n\geq 1$, then it follows from the
definition that $\mathcal{E}(\omega^{*})=-\mathcal{E}(\omega)$. A
straightforward induction shows that
$\mathfrak{p}(\omega^{*})=-\mathfrak{p}(\omega)$ and
$\mathfrak{q}(\omega^{*})=\mathfrak{q}(\omega)$, hence
$\mu_{\omega^{*}}(z)=-\mu_{\omega}(-z)$. ∎

It will be convenient to split a modified continued fraction into smaller
pieces. For any $n\in\mathbb{Z}_{\geq 0}\cup\\{\infty\\}$, integer $0\leq
m\leq n$, and $\omega=\langle
a_{j}:\varepsilon_{j}\rangle_{j=1}^{n}\in\mathcal{Q}_{n}$, we define

$\displaystyle\lfloor\omega\rfloor_{m}$ $\displaystyle:=\langle
a_{j}:\varepsilon_{j}\rangle_{j=1}^{m}\in\mathcal{Q}_{m},\text{ and}$
$\displaystyle\lceil\omega\rceil^{m}$ $\displaystyle:=\langle
a_{j}:\varepsilon_{j}\rangle_{j=m+1}^{n}\in\mathcal{Q}_{n-m}.$

This decomposition of modified continued fractions induces a decomposition of
their associated Möbius transformations:

######  Proposition 1.3.

If $\omega\in\mathcal{Q}_{n}$ for some $n\in\mathbb{Z}_{\geq
0}\cup\\{\infty\\}$, then

$\mu_{\omega}(z)=\mu_{\lfloor\omega\rfloor_{m}}\left(\mathcal{E}(\lfloor\omega\rfloor_{m})\mu_{\lceil\omega\rceil^{m}}(\mathcal{E}(\omega)\mathcal{E}(\lceil\omega\rceil^{m})z)\right)$

for all $0\leq m\leq n$ and $z\in\hat{\mathbb{C}}$.

###### Proof.

When $n\neq\infty$, this follows directly from (1) and

$\mu_{\omega}(z)=\cfrac{\varepsilon_{1}}{a_{1}+\cfrac{\varepsilon_{2}}{\ddots+\cfrac{\varepsilon_{m}}{a_{m}+\mathcal{E}(\lfloor\omega\rfloor_{m})^{2}\cdot\cfrac{\varepsilon_{m+1}}{a_{m+1}+\cfrac{\varepsilon_{m+2}}{\ddots+\cfrac{\varepsilon_{n}}{a_{n}+\mathcal{E}(\lceil\omega\rceil^{m})^{2}\mathcal{E}(\omega)z}}}}}}.$

The $n=\infty$ case holds similarly. ∎

Finally, let us observe the following geometric property of the Möbius
transformations $\mu_{\omega}$:

######  Proposition 1.4.

For any $\omega=\langle
a_{j}:\varepsilon_{j}\rangle_{j=1}^{n}\in\mathcal{Q}_{n}$ with $n\geq 1$, if
$|z|\leq 1$ then

$|\mu_{\omega}(z)|,|\arg(\epsilon_{1}\mu_{\omega}(z))|\leq\frac{1}{a_{1}-1}.$

###### Proof.

As $a_{1}\geq 2$ and

$\mu_{\omega}(z)=\frac{\varepsilon_{1}}{a_{1}+\mu_{\lceil\omega\rceil^{1}}(\mathcal{E}(\omega)\mathcal{E}(\lceil\omega\rceil^{1})z)}$

by proposition 1.3, the desired result follows from a straightforward
induction on $n$. ∎

##  2\. Parabolic implosion

Given a holomorphic function $h$ defined on a region $U$, we will say that $U$
is a petal for $h$ if:

  1. (1)

$U$ is an open Jordan domain.

  2. (2)

There exists a univalent map $\phi:U\to\mathbb{C}$, called a Fatou coordinate,
satisfying:

    1. (a)

For all $z\in U$, the following are equivalent:

       * •

$h(z)\in U$.

       * •

$\phi(z)+1\in\phi(U)$.

       * •

$\phi(h(z))=\phi(z)+1$.

    2. (b)

If both $w$ and $w+n$ belong to $\phi(U)$ for some $n\geq 0$, then
$w+j\in\phi(U)$ for all $0\leq j\leq n$.

    3. (c)

For any $w\in\mathbb{C}$, there exists $n\in\mathbb{Z}$ such that
$w+n\in\phi(U)$.

If $h(U)\subset U$ or $U\subset h(U)$, then we will say that $U$ is an
attracting or repelling petal respectively for $h$. The Fatou coordinate for a
petal is unique up to post-composition by a translation, see for example
[Mil06] or [Shi00].

Let us fix some $\omega\in\mathcal{Q}$ and set
$p=\mathfrak{p}(\omega),q=\mathfrak{q}(\omega)$. We also fix an analytic
function $f$ defined on a neighborhood of $0$ such that

$f(z)=e^{2\pi i\mu_{\omega}(0)}z+O(z^{2})$

and

$f^{q}(z)=z+az^{q+1}+O(z^{q+2})$

near $z=0$ for some $a\neq 0$. In this case we will say that $f$ has a non-
degenerate $\omega$-parabolic fixed point at $0$. Throughout this paper we
will consider analytic maps in the compact-open topology, i.e. a neighborhood
of $f$ is a set of the form

$\left\\{h:Dom(h)\to\mathbb{C}\;\middle|\;\begin{tabular}[]{@{}l@{}}$h\text{
is analytic on }Dom(h)\supset K,\text{ and }$\\\ $|f(z)-h(z)|<\epsilon\text{
for all }z\in K$\end{tabular}\right\\}$

for some $\epsilon>0$ and compact set $K\subset Dom(f).$

Let $h$ be another analytic map defined on a neighborhood of $0$ such that

$h(z)=e^{2\pi i\mu_{\omega}(\alpha)}z+O(z^{2})$

near $z=0$ for some $\alpha\in\mathbb{C}$. If $\alpha=0$ and $h$ is
sufficiently close to $f$, then $h$ also has a non-degenerate
$\omega$-parabolic fixed point at $0$. It is a classical result of Leau
[Lea97] and Fatou [Fat20] that in this case, the local dynamics of $h^{q}$ can
be completely described by petals:

######  Theorem 2.1 (Leau, Fatou).

If $\alpha=0$ and if $h$ is sufficiently close to $f$, then in any
neighborhood of $z=0$ there exist pair-wise disjoint petals
$P_{j}=P_{h,j}^{f}$ for $h^{q}$, where the subscript $j$ ranges over
$\mathbb{Z}/2q\mathbb{Z}$, such that:

  1. (1)

For all $j$, $P_{j}$ is bounded by a simple closed curve containing $0$.

  2. (2)

The petal $P_{j}$ is attracting (resp. repelling) if $j$ is even (resp. odd).

  3. (3)

The counter-clockwise circular ordering of the sets $P_{j}$ about $z=0$ is
induced exactly by the ordering of $\mathbb{Z}/2q\mathbb{Z}$ (see Figure 1).

  4. (4)

Every forward (resp. backwards) orbit under $h^{q}$ which converges
asymptotically towards $z=0$ eventually intersects $P_{j}$ for some even
(resp. odd) $j$.

###### Proof.

See [Mil06]. ∎

We will call the petals $P_{j}$ as in theorem 2.1 petals for $h$ relative to
$f$. Here the “relative to $f$” distinction is not important as the petals do
not depend on $f$; we include this description to keep our notation consistent
when we consider the $\alpha\neq 0$ case below.

While the petals $P_{j}$ describe the local dynamics of $h^{q}$, it is
important to also consider the action of $h$:

######  Proposition 2.2.

For any even $j$ and $z\in P_{j}$, $h^{nq+1}(z)\in P_{j+2p}$ for all large
$n$. For any odd $j$ and $z\in P_{j}$, there exists some $z^{\prime}\in
P_{j-2p}$ such that $h^{nq+1}(z^{\prime})=z$ and $h^{mq+1}(z^{\prime})\in
P_{j}$ for all $0\leq m\leq n$.

###### Proof.

This follows from the circular ordering of the sets $P_{j}$ about $z=0$ and
the fact that $h(z)\approx e^{2\pi ip/q}z$ near $z=0$. ∎

Now let us consider the case where $\alpha\neq 0$. For all $r>0$, we denote

$A_{r}^{+}=\\{\alpha\in\mathbb{C}^{*}:|\alpha|<r\text{ and
}|\arg\,\alpha|<\pi/4\\}.$

The fundamental phenomenon in parabolic implosion is that if $\alpha\in
A_{1}^{+}$ and $h$ is sufficiently close to $f$, then we can still define
petals near $z=0$:

$P_{0}$
---
$P_{1}$
---
$P_{2}$
---
$P_{3}$
---
$P_{4}$
---
$P_{5}$
---
$P_{0}=P_{1}$
---
$P_{2}=P_{3}$
---
$P_{4}=P_{5}$
---
Figure 1. Left: Petals for $h$ relative to $f$ as in theorem 2.1 with
$\alpha=0$ and $q=3$. Right: Petals for $h$ relative to $f$ as in proposition
2.3 with $\alpha\in A_{1}^{+}$ and $q=3$.

######  Proposition 2.3.

If $\alpha\in A_{1}^{+}$ and if $h$ is sufficiently close to $f$, then in any
neighborhood of $z=0$ there exist pair-wise disjoint petals

$P_{j}=P_{h,j}^{f}=P_{j+1}=P_{h,j+1}^{f}$

for $h^{q}$, where the subscript $j$ ranges over $2\mathbb{Z}/2q\mathbb{Z}$,
such that:

  1. (1)

For all $j$, $P_{j}$ is bounded by two curves connecting $0$ to a non-zero
fixed point of $h^{q}$.

  2. (2)

The counter-clockwise circular ordering of the petals $P_{j}$ about $z=0$ is
induced exactly by the ordering of $2\mathbb{Z}/2q\mathbb{Z}$ (see Figure 1).

  3. (3)

For all $j$, let $\phi_{j}=\phi_{h,j}^{f}$ be a Fatou coordinate on $P_{j}$.
For $z\in P_{j}$, $z\to 0$ when $\emph{Im}\,\phi_{j}(z)\to+\infty$.

We will call the petals $P_{j}$ as in proposition 2.3 petals for $h$ relative
to $f$. In this setting, the “relative to $f$” distinction is very important:
$h$ could also approximate a different analytic function which has a non-
degenerate $\omega^{\prime}$-parabolic fixed point for some
$\omega^{\prime}\neq\omega$, in this case we could define completely different
petals for $h$.

In both the $\alpha=0$ and $\alpha\in A_{1}^{+}$ cases, the petals $P_{j}$ are
not canonical. It will be convenient to restrict our attention to petals which
have additional geometric structure.

######  Proposition 2.4.

There exist constants $M,r>0$ which depend only on $f$ such that if $h$ is
sufficiently close to $f$ and $\alpha\in A_{r}^{+}\cup\\{0\\}$, then for any
$\theta\in[-\frac{\pi}{13},\frac{\pi}{13}]$ the petals $P_{j}$ and Fatou
coordinates $\phi_{j}$ may be chose so that:

  1. (1)

If $j$ is even then

$\phi_{j}(P_{j})=\left\\{w\in\mathbb{C}\;\middle|\;\begin{tabular}[]{@{}l@{}}$\emph{Re}\,w>\emph{Im}(w)\tan\theta\text{
and }$\\\
$\emph{Re}(w-\frac{1}{\alpha})<-M+\emph{Im}(w-\frac{1}{\alpha})\tan\theta$\end{tabular}\right\\},$

and if $j$ is odd then

$\phi_{j}(P_{j})=\left\\{w\in\mathbb{C}\;\middle|\;\begin{tabular}[]{@{}l@{}}$\emph{Re}\,w<\emph{Im}(w)\tan\theta\text{
and }$\\\
$\emph{Re}(w+\frac{1}{\alpha})>M+\emph{Im}(w-\frac{1}{\alpha})\tan\theta$\end{tabular}\right\\},$

using the convention that $\infty<\infty$ (see Figure 2).

  2. (2)

The compact sets $\overline{P_{j}\cup P_{j+1}}$ and
$\hat{\mathbb{C}}\setminus(P_{j}\cup P_{j+1})$ depend continuously on $h$ and
$\theta$ for all $j\in 2\mathbb{Z}/2q\mathbb{Z}$. Here we use the Hausdorff
topology on non-empty compact subsets of $\mathbb{C}$.

###### Proof.

This fact is proved in [Oud02]. ∎

$\theta$
---
$\phi_{0}$
---
$\mathbb{C}$
---
$0$
---
$-M+\frac{1}{\alpha}$
---
$\theta$
---
$P_{0}$
---
$\phi_{0}$
---
$\mathbb{C}$
---
$0$
---
$P_{1}$
---
$P_{0}=P_{1}$
---
Figure 2. Left: Petals and Fatou coordinates for $h$ relative to $f$ as in
proposition 2.4 with Fatou-tilt $\theta$, $\alpha=0$, and $q=1$. Right: Petals
and Fatou coordinates for $h$ relative to $f$ as in proposition 2.4 with
Fatou-tilt $\theta$, $\alpha\in A_{1}^{+}$, and $q=1$.

For the rest of this paper, we will assume that the petals $P_{j}$ are chosen
so that they satisfy proposition 2.4 for some $\theta$, we will call the angle
$\theta$ the Fatou-tilt of the petals. However, we will usually modify the
Fatou coordinates by a translation; preferred choices of Fatou coordinates are
described later in this section.

While proposition 2.4 fixes the images of the petals $P_{j}$ under the
respective Fatou coordinates, it does not fix a choice of Fatou coordinates.
We will usually fix the Fatou coordinates $\phi_{j}$ with $j\in
2\mathbb{Z}/2q\mathbb{Z}$ by picking a point $z_{h}\in P_{j}$ which depends
continuously and analytically on $h$ and setting $\phi_{j}(z_{h})=w$ for some
$w\in\mathbb{C}$ which does not depend on $h$. In this case, we will say that
the Fatou coordinates $\phi_{j}$ with $j$ even are normalized. For the rest of
this paper we will assume that these Fatou coordinates are normalized.

######  Proposition 2.5.

The normalized Fatou coordinates $\phi_{j}$ with $j$ even depends continuously
and holomorphically on $h$ near $f$.

For $j$ odd, we will make a different choice for the Fatou coordinate
$\phi_{j}$ which is specified later in this section.

###  2.1. Renormalizations

Let us keep the same notation as above. For all $j$ and $z\in P_{j}$, we will
say that $z$ is petal-entering if $j$ is even and

$\phi_{j}(z)-\text{Re}\frac{1}{4\alpha}\notin\overline{\phi_{j}(P_{j})}\subset\mathbb{C},$

and say that $z$ is petal-exiting if $j$ is odd and

$\phi_{j}(z)+\text{Re}\frac{1}{4\alpha}\notin\overline{\phi_{j}(P_{j})}\subset\mathbb{C}.$

When $\alpha=0$, every point in $P_{j}$ is petal-entering or petal-exiting if
$j$ is even or odd respectively. When $\alpha\in A_{1}^{+}$, so
$P_{j}=P_{j+1}$ for even $j$, then there are both petal-entering and petal-
exiting points in $P_{j}$. The following lemma keeps these points separated:

######  Lemma 2.6.

Assume $\alpha\in A_{1}^{+}$. If $h$ is sufficiently close to $f$, then for
any $j$, petal-entering $z\in P_{j}$, and petal-exiting $z^{\prime}\in P_{j}$
such that $\phi_{j}(z)-\phi_{j}(z^{\prime})\in\mathbb{R},$ we have

$\emph{Re}\frac{1}{4\alpha}<\phi_{j}(z^{\prime})-\phi_{j}(z)<\emph{Re}\frac{5}{4\alpha}.$

###### Proof.

Proposition 2.4 implies that $\phi_{j}(P_{j})$ is bounded by two lines,

$\ell\text{ and
}\ell+\text{Re}({1}/{\alpha})-M-\text{Im}({1}/{\alpha})\tan\theta,$

where $\theta$ is the Fatou-tilt of the petals. As $|\arg\,\alpha|<\pi/4$,
$|\arg\,\theta|<\pi/13$, $1-\tan(\frac{\pi}{13})>3/4$, and
$1+\tan(\frac{\pi}{13})<5/4$, if $\alpha$ is sufficiently then

$\text{Re}\frac{1}{4\alpha}<\text{Re}\frac{1}{\alpha}-M-(\tan\theta)\text{Im}\frac{1}{\alpha}<\text{Re}\frac{5}{4\alpha}.$

∎

Fix some odd $\ell\in\mathbb{Z}/2q\mathbb{Z}$ and let $z\in P_{\ell}$ be
petal-exiting and assume that:

  1. (1)

There exists some integers $0\leq m<\text{Re}\frac{1}{4\alpha}$ and $0\leq
k<q$ such that $h^{mq+k}(z)$ is petal-entering in $P_{\ell-1}$.

  2. (2)

There exists a minimal integer $0<m^{\prime}\leq m$ such that $h^{mq}$ is
petal-entering in $P_{j}$ for some even $j\in\mathbb{Z}/2q\mathbb{Z}$.

  3. (3)

$j+2pk\equiv\ell-1\mod 2q$.

In this case, we define

$H_{h,\ell}^{f}(\phi_{\ell}(z))=\phi_{\ell-1}(h^{mq+k}(z))-m.$

We will call $H_{\ell}=H_{h,\ell}^{f}$ a horn map for $h$ relative to $f$.

######  Proposition 2.7.

If $h$ is sufficiently close to $f$, then $H_{\ell}$ is well-defined and
analytic for all odd $\ell$. Moreover,

$H_{\ell}(w+j)=H_{\ell}(w)+j$

for any $w\in\mathbb{C}$ and $j\in\mathbb{Z}$ where both sides of the equation
are defined and

###### Proof.

The horn map $H_{\ell}$ is analytic wherever it is well-defined as
$\phi_{\ell-1},\phi_{\ell}^{-1},$ and $h$ are analytic. Fix some
$w\in\mathbb{C}$ and $j\in\mathbb{Z}$ such that both $H_{\ell}(w)$ and
$H_{\ell}(w+j)$ are defined. Set

$z_{1}=(\phi_{\ell})^{-1}(w)\text{ and }z_{2}=(\phi_{\ell})^{-1}(w+j),$

without loss of generality we can assume that $j\geq 0$, so
$z_{2}=h_{\alpha}^{jq}(z_{1})$. By definition, both $z_{1}$ and $z_{2}$ are
petal-exiting, hence $j<\text{Re}\frac{1}{4\alpha}.$

The definition of the horn map $H_{\ell}$ implies that there exist integers
$0\leq m_{1},m_{2}<\text{Re}\frac{1}{4\alpha}$ and $0\leq k_{1},k_{2}<q$ and
such that $h^{m_{s}q+k_{s}}(z_{s})$ is petal-entering in $P_{\ell-1}$ for
$s=1,2$. Moreover, for $s=1,2$ there exists an even integer $j_{s}$ and a
minimal integer $0<m_{s}^{\prime}\leq m_{s}$ such that $h^{m_{s}q}(z_{s})$ is
petal-entering in $P_{j_{s}}$. We assume that $m_{2}^{\prime}\leq
m_{1}^{\prime}$, the argument when $m_{2}^{\prime}>m_{1}^{\prime}$ follows
similarly. Thus

$P_{j_{1}}\ni h^{jq}(h^{m_{1}q}(z_{1}))=h^{(m_{1}-m_{2})q}(h^{m_{2}q}(z))\in
P_{j_{2}}$

as $j$ and $m_{1}-m_{2}$ are both smaller than $\text{Re}\frac{1}{4\alpha}$.
Hence $j_{1}=j_{2}$ as the petals $P_{0},P_{2},\dots,P_{2q-2}$ are all
disjoint, so $k_{1}=k_{2}$. Thus

$\displaystyle H_{\ell}(w+j)$
$\displaystyle=\phi_{\ell-1}(h^{m_{2}q+k_{2}}(z_{2}))-m_{2}$
$\displaystyle=\phi_{\ell-1}(h^{m_{1}q+k_{2}}(z_{2}))-m_{1}$
$\displaystyle=\phi_{\ell-1}(h^{(m_{1}+j)q+k_{1}}(z_{1}))-m_{1}$
$\displaystyle=\phi_{\ell-1}(h^{m_{1}q+k_{1}}(z_{1}))+j-m_{1}$
$\displaystyle=H_{\ell}(w)+j.$

∎

Using proposition 2.7, for all odd $\ell$ we can analytically extend
$H_{\ell}$ to a domain invariant under the translation $z\mapsto z+1$ such
that

$H_{\ell}(w+1)=H_{\ell}(w)+1.$

######  Proposition 2.8.

For any odd $\ell$, there exists $\eta>0$ such that the domain of $H_{\ell}$
contains $\\{w\in\mathbb{C}:|\emph{Im}\,w|>\eta\\}$. Moreover, there exist
constants $\zeta_{-},\zeta_{+}\in\mathbb{C}$ such that
$H_{\ell}(w)-w\to\zeta_{\pm}$ when $\emph{Im}\,w\to\pm\infty$.

For any normalization of $\phi_{\ell-1}$, there exists a unique choice of
$\phi_{\ell}$ so that $\zeta_{+}=\frac{1-\mathcal{E}(\omega)}{2}$. With this
choice, the Fatou coordinate $\phi_{\ell}$ and horn map $H_{\ell}$ depends
continuously and holomorphically on $h$ near $f$.

###### Proof.

The case where $\omega=\emptyset$ is proved in [Shi00]. The same argument
holds for the general $\omega$ case. ∎

We will say that the Fatou coordinates $\phi_{j}$ with $j$ odd are normalized
if they are chosen as in proposition 2.8. This choice of Fatou coordinates is
made so that we can explicitly compute the difference between $\phi_{j}$ and
$\phi_{j+1}$ for even $j$ when $\alpha\neq 0:$

######  Proposition 2.9.

For $h$ sufficiently close to $f$ and $\alpha\in A_{1}^{+}$, the normalized
Fatou coordinates $\phi_{j}$ and $\phi_{j+1}$ satisfy

$\phi_{j}(z)-\frac{1}{\alpha}=\phi_{j+1}(z)$

for all $z\in P_{j}=P_{j+1}$ and even $j$.

###### Proof.

Without loss of generality we will only consider $j=0$, the argument for other
choices of $j$ is identical. Assume that the choice of $\phi_{0}$ is given and
set $\phi_{1}(z)=\phi_{0}(w)-1/\alpha$; to prove the proposition it suffices
to show that $H_{1}(w)-w\to\frac{1-\mathcal{E}(\omega)}{2}$ when
$\text{Im}\,w\to+\infty$. Note that $H_{1}$ depends only on $\phi_{0}$ and
$\phi_{1}$ so we are free to change the other Fatou coordinates.

For all $j\in\mathbb{Z}/2q\mathbb{Z}$, let $S_{j}$ be the set of petal-
entering (resp. petal-exiting) points in $P_{j}$ when $j$ is even (resp. odd).
It follows from proposition 2.3 that counter-clockwise cyclic ordering about
$z=0$ of $S_{j}$ with $j\in\mathbb{Z}/2q\mathbb{Z}$ is given by the ordering
on $\mathbb{Z}/2q\mathbb{Z}.$ Moreover, as $(h^{q})^{\prime}(0)\approx e^{2\pi
i\alpha/q}$ and $\arg\alpha\in(-\frac{\pi}{4},\frac{\pi}{4})$, the orbit under
$h^{q}$ of any point close to $0$ must pass through the regions $S_{j}$
according to this cyclic ordering.

Let $\tilde{H}$ be the map induced by the first return map to $S_{1}$ under
$h^{q}$, i.e. if $z\in S_{1}$, $w=\phi_{1}(z)$, and $m>1$ is the minimal
integer such that $h^{mq}(z)\in S_{1}$, then

$\tilde{H}(w):=\phi_{1}(h^{mq}(z))-m.$

It is proved in [Oud02] that

$\lim_{\text{Im}\,w\to\infty}\tilde{H}(w)-w=-\frac{2\pi
i}{\log(h^{q})^{\prime}(0)}=\frac{-1}{q\mu_{\omega}(\alpha)-p}=-\frac{q}{\alpha}-\mathcal{E}(\omega)\mathfrak{q}_{n-1}(\omega).$

For all $j\in\mathbb{Z}/2q\mathbb{Z}$, let $\tilde{H}_{j}$ be the map induced
by iterating points in $S_{j}$ until they enter $S_{j+1}$. More precisely, if
$z\in S_{j}$, $w=\phi_{j}(z)$, and $m>1$ is the minimal integer such that
$h^{mq}(z)\in S_{j+1}$, then

$\tilde{H}_{j}(w):=\phi_{j+1}(h^{mq}(z))-m.$

Thus when $\text{Im}\,w$ is sufficiently large,

$\tilde{H}(w)=\tilde{H}_{2q}\circ\dots\circ\tilde{H}_{1}(w).$

For all even $j$ we choose the Fatou coordinates so that
$\phi_{j+1}(z)=\phi_{j}(z)-1/\alpha$, thus $\tilde{H}_{j}(w)=w-1/\alpha$.
Similarly to proposition 2.8, arguments in [Shi00] and [Oud02] imply that for
all odd $j$ there exists $\zeta_{j}\in\mathbb{C}$ such that
$\tilde{H}_{j}(w)-w\to\zeta_{j}$ when $\text{Im}\,w\to+\infty.$ Hence

$-\frac{q}{\alpha}-\mathcal{E}(\omega)\mathfrak{q}_{n-1}(\omega)=-\frac{q}{\alpha}+\sum_{k=0}^{q-1}\zeta_{1+2k}.$

We now fix some $z_{1}\in S_{1}$, set $w=\phi_{1}(z_{1})$ and assume that
$\tilde{H}_{1}(w)$ is defined. Thus there exists some minimal integer $m\geq
0$ such that

$z_{2}:=h^{mq}(z_{1})\in S_{2}.$

Keeping $w$ fixed, it follows from proposition 2.2 that if $h$ is sufficiently
close to $f$ then there exist minimal integers $m_{2}\geq 0$ and $0\leq
k_{2}<q$ such that

$z_{0}:=h^{m_{2}q+k_{2}}(z_{2})\in S_{0}.$

Similarly, if $h$ is sufficiently close to $f$ then for all
$j\in\mathbb{Z}/2q\mathbb{Z}$ there exists a point $z_{j}\in S_{j}$ and
minimal integers $m_{j}>0$, $0\leq k_{j}<q$ such that

$h^{m_{j}q+k_{j}}(z_{j})=\begin{cases}z_{0}&\text{ if }j\text{ is even}\\\
z_{1}&\text{ if }j\text{ is odd}\end{cases}.$

Moreover, $k_{j}$ satisfies

(2) $j+2pk_{j}\equiv\begin{cases}0\mod 2q&\text{ if }j\text{ is even}\\\ 1\mod
2q&\text{ if }j\text{ is odd}\end{cases}.$

Changing the choice of Fatou coordinates, we can assume without loss of
generality that $\phi_{j}(z_{j})=\phi_{1}(z_{1})-m_{j}$ for all odd $j$. Thus
$\phi_{j}(z_{j})$ is induced exactly by iterating $z_{j}$ until it enters
$P_{1}$ for all even $j$. If $h$ is sufficiently close to $f$ and if $z_{1}$
is sufficiently close to $0$, then $\phi_{j}(z_{j})$ is similarly induced by
iterating $z_{j}$ until it enters $P_{1}=P_{0}$, so
$\phi_{j}(z_{j})=\phi_{0}(z_{0})-m_{j}$, for all even $j$.

As $h$ is invertible near $z=0$, if $z_{1}$ is sufficiently close to $0$ then

$\displaystyle h^{(m+m_{j}+m_{2}-m_{j+1})q+k_{j}+k_{2}-k_{j+1}}(z_{j})$
$\displaystyle=h^{(m+m_{2}-m_{j+1})q+k_{2}-k_{j+1}}(z_{1})$
$\displaystyle=h^{(m_{2}-m_{j+1})q+k_{2}-k_{j+1}}(z_{2})$
$\displaystyle=h^{-m_{j+1}q-k_{j+1}}(z_{0})$ $\displaystyle=z_{j+1}$

for all odd $j$. The definition of $\tilde{H}_{j}$ therefore implies

$\displaystyle\tilde{H}_{j}(w)$ $\displaystyle=\tilde{H}_{j}(w-m_{j})+m_{j}$
$\displaystyle=\phi_{j+1}(z_{j+1})-m-m_{2}+m_{j+1}-\left\lfloor\frac{k_{j}+k_{2}-k_{j+1}}{q}\right\rfloor$
$\displaystyle=\phi_{0}(z_{0})-m-m_{2}-\left\lfloor\frac{k_{j}+k_{2}-k_{j+1}}{q}\right\rfloor$
$\displaystyle=\phi_{2}(z_{2})-m-\left\lfloor\frac{k_{j}+k_{2}-k_{j+1}}{q}\right\rfloor$
$\displaystyle=\tilde{H}_{1}(w)-\left\lfloor\frac{k_{j}+k_{2}-k_{j+1}}{q}\right\rfloor.$

By (2), we must have $k_{j}+k_{2}-k_{j+1}\equiv 0\mod q$, thus

$\tilde{H}_{j}(w)=\begin{cases}\tilde{H}_{1}(w)&\text{if }k_{j+1}\geq k_{2}\\\
\tilde{H}_{1}(w)-1&\text{if }k_{j+1}<k_{2}\end{cases}$

as $k_{j}=0$ if and only if $j=0,1$.

Proposition 1.1 implies that

$k_{2}=\frac{1-\mathcal{E}(\omega)}{2}q+\mathcal{E}(\omega)\mathfrak{q}_{n-1}(\omega)=\begin{cases}\mathfrak{q}_{n-1}(\omega)&\text{
if }\mathcal{E}(\omega)=+1\\\ q-\mathfrak{q}_{n-1}(\omega)&\text{ if
}\mathcal{E}(\omega)=-1\end{cases}.$

As the integers $k_{j}$ range over the integers $0,\dots,q-1$ when $j$ ranges
over the even integers in $\mathbb{Z}/2q\mathbb{Z}$, there are exactly $k_{2}$
choices of even $j\in\mathbb{Z}/2q\mathbb{Z}$ such that $k_{j}<k_{2}$. Thus

$-\mathcal{E}(\omega)\mathfrak{q}_{n-1}(\omega)=\sum_{k=0}^{q-1}\zeta_{1+2k}=q\zeta_{1}-k_{2},$

so $\zeta_{1}=\frac{1-\mathcal{E}(\omega)}{2}.$ Moreover, for $w$ as above we
have

$h^{(m+m_{2})q-k_{2}}(z_{1})=z_{0}.$

The definition of the horn map implies

$H_{1}(w)=\phi_{0}(z_{0})-m-m_{2}=\phi_{2}(z_{2})-m=\tilde{H}_{1}(w),$

hence $H_{1}(w)-w\to\zeta_{1}$ when $\text{Im}\,w\to+\infty$. ∎

Let us remark here that proposition 2.9 allows us to recover a generalized
version of Lavaurs’ theorem from [Lav89] and [Dou94]:

######  Theorem 2.10.

Let $h_{n}$ be a sequence of analytic functions converging locally uniformly
to $f$ with $h_{n}^{\prime}(0)=e^{2\pi i\mu_{\omega}(\alpha_{n})}$. If there
exists a sequence $k_{n}\to+\infty$ and some $\delta\in\mathbb{C}$ such that
$k_{n}-1/\alpha_{n}\to\delta$, then for any even $\ell$, $z_{0}\in
P_{f,\ell}^{f}$, and sufficiently large integer $j\geq 0$,

$h_{n}^{(k_{n}-j)q}(z)\to(\phi^{f}_{f,\ell+1})^{-1}(\phi_{f,\ell}^{f}(z)+\delta-j)$

locally uniformly in a neighborhood of $z_{0}$.

###### Proof.

Proposition 2.9 implies that for any even $\ell$, $z_{0}\in P_{f,\ell-1}^{f}$,
and sufficiently large integer $j$,

$h_{n}^{(k_{n}-j)q}(z)=(\phi_{h_{n},\ell+1}^{f})^{-1}(\phi_{h_{n},\ell}^{f}(z)+k_{n}-j-1/\alpha_{n})$

for all $z$ close to $z_{0}$ and all $n$ sufficiently large. The theorem
therefore follows from the convergence
$\phi_{h_{n},\ell}^{f}\to\phi_{f,\ell}^{f}$ for all $\ell$. ∎

While the horn maps $H_{1},H_{3},\dots,H_{2q-1}$ are all defined individually,
we can relate these different maps using proposition 2.2. It usually suffices
to consider just a single horn map, so we will denote
$H=H_{h}^{f}:=H_{h,1}^{f}$.

Setting $\text{Exp}(z):=e^{2\pi iz}$, we define the (top) near-parabolic
renormalization of $h$ relative to $f$ to be the map

$\mathcal{R}_{f}h:=\text{Exp}\circ H_{h}^{f}\circ\text{Exp}^{-1}.$

Proposition 2.8 implies that $\mathcal{R}_{f}h$ is defined and analytic on
punctured neighborhoods of $z=0$ and $z=\infty$, and we can analytically
extend $\mathcal{R}_{f}h(z)$ by setting
$\mathcal{R}_{f}h(0)=\mathcal{R}_{f}h(0)$ and
$\mathcal{R}_{f}h(\infty)=\mathcal{R}_{f}h(\infty)$. Moreover, proposition 2.8
implies that $(\mathcal{R}_{f}h)^{\prime}(0)=1$ and $\mathcal{R}_{f}h$ depends
continuously and analytically on $h$. We will also call $\mathcal{R}_{f}h$ the
(top) parabolic renormalization of $h$ relative to $f$ when $\alpha=0$ to
distinguish from the $\alpha\neq 0$ case.

We define the bottom near-parabolic renormalization of $h$ relative to $f$ to
be the map defined by

$\mathcal{R}_{f}^{\flat}h(z)=\frac{\lambda}{\mathcal{R}_{f}h(1/z)},$

where $\lambda\in\mathbb{C}^{*}$ is chosen so that
$(\mathcal{R}_{f}^{\flat}h)^{\prime}(0)=1$. The top and bottom near-parabolic
renormalizations of $h$ are closely related; in this paper we will usually
focus on the top renormalization, but the bottom renormalization can be
considered by symmetric reasoning. Just as for the top renormalization, we
will also call $\mathcal{R}_{f}^{\flat}h$ the bottom parabolic renormalization
of $h$ relative to $f$ when $\alpha=0$.

A key feature of the near-parabolic renormalization $\mathcal{R}_{f}h$ is its
relationship to large iterates of $h$ when $\alpha\neq 0$:

######  Proposition 2.11.

Assume $\alpha\in A_{1}^{+}$. If $h$ is sufficiently close to $f$, then for
any $n\geq 1$ and petal-exiting $z_{1},z_{2}\in P_{1}$ such that

(3) $(e^{-2\pi
i/{\alpha}}\mathcal{R}_{f}h)^{n}\circ\emph{Exp}\circ\phi_{1}(z_{1})=\emph{Exp}\circ\phi_{1}(z_{2})$

there exist integers
$\emph{Re}(\frac{n}{4\alpha})<m<\emph{Re}(\frac{7n}{4\alpha})$ and $0\leq
k<nq$ such that

$h^{mq+k}(z_{1})=z_{2}.$

###### Proof.

First we consider the case $n=1$. Set $w_{1}=\phi_{1}(z_{1})$ and
$w_{2}=\phi_{1}(z_{2})$. The definition of $\mathcal{R}_{f}h$ and (3) imply
that there is an integer $j$ such that

$H(w_{1})-1/\alpha=w_{2}-j.$

The definition of $H$ implies that there is some petal-exiting $z^{\prime}\in
P_{1}$ such that:

  1. (1)

There is an integer $m^{\prime}$ such that
$\phi_{1}(z^{\prime})=w_{1}+m^{\prime}.$

  2. (2)

There exist integers $0\leq m<\text{Re}\frac{1}{4\alpha}$ and $0\leq k<q$ such
that $h^{mq+k}(z^{\prime})\in P_{0}$ is petal-entering.

  3. (3)

$H(w_{1}+m^{\prime})=\phi_{0}(h^{mq+k}(z^{\prime}))-m.$

Thus

$\displaystyle\phi_{0}(z_{2})-j=w_{2}+1/\alpha-j=H(w_{1}+m^{\prime})=\phi_{0}(h^{mq+k}(z^{\prime}))-m.$

As both $z_{1}$ and $z^{\prime}$ are petal-exiting, we must have
$|m^{\prime}|<\text{Re}\frac{1}{4\alpha}$. If $m^{\prime}<0$ then we could
replace $z^{\prime}$ with $z_{1}$, so we assume that $m^{\prime}\geq 0$.

As $z_{2}$ is petal-exiting and $h^{mq+k}(z^{\prime})$ is petal-entering,
lemma 2.6 implies that

$\text{Re}\frac{1}{4\alpha}<j-m<\text{Re}\frac{5}{4\alpha}.$

Thus

$z_{2}=h^{(j-m)q}\circ
h^{mq+k}(z^{\prime})=h^{jq+k}(z^{\prime})=h^{(j+m^{\prime})q+k}(z_{1})$

and $\text{Re}\frac{1}{4\alpha}<j+m^{\prime}<\text{Re}\frac{7}{4\alpha}.$

Now we assume that $n>1$. Repeated application of the $n=1$ case yields the
desired result. ∎

While so far in this section we have only considered perturbations $h$ of $f$
with $\alpha\in A_{1}^{+}\cup\\{0\\}$, we can perform a similar construction
when $\alpha\in A_{r}^{-}:=-A_{r}^{+}.$ Let us denote by $f^{*}$ and $h^{*}$
the functions $\overline{f(\overline{z})}$ and $\overline{h(\overline{z})}$
respectively, so

$(h^{*})^{\prime}(0)=e^{-2\pi i\mu_{\omega}(\alpha)}=e^{2\pi
i\mu_{\omega^{*}}(-\alpha)}.$

Similarly, $f^{*}$ has a non-degenerate $\omega^{*}$-parabolic fixed point at
$z=0$. Viewing $h^{*}$ as a perturbation of $f^{*}$, we define the top near-
parabolic renormalization of $h$ relative to $f$ to be the map

$\mathcal{R}_{f}h:=\mathcal{R}_{f^{*}}h^{*}$

when $\alpha\in A_{1}^{-}$. We can similarly define the bottom renormalization
by

$\mathcal{R}^{\flat}_{f}h:=\mathcal{R}^{\flat}_{f^{*}}h^{*}.$

Setting $A_{r}=A_{r}^{+}\cup A_{r}^{-}$, the operators $\mathcal{R}_{f}$ and
$\mathcal{R}_{f}^{\flat}$ are therefore defined on all $h$ sufficiently close
to $f$ with $\alpha\in A_{1}\cup\\{0\\}$.

Unfortunately, the operator $\mathcal{R}_{f}$ is not well-defined in general
as $\mathcal{R}_{f}h$ depends on the choice of petals $P_{j}$ and Fatou
coordinates $\phi_{j}$. Changing the petals $P_{j}$ will change the domain of
definition of $\mathcal{R}_{f}h$, and changing the Fatou coordinates will
conjugate $\mathcal{R}_{f}h$ by a linear map. In the most general setting,
$\mathcal{R}_{f}$ is well-defined as an operator acting on analytic germs near
$z=0$ up to linear conjugacy. The same is true for the bottom renormalization.
In the next section we will restrict to classes of maps for which we can
canonically choose the petals and Fatou coordinates, so the corresponding
near-parabolic renormalizations will be well-defined as operators on functions
instead of germs.

##  3\. Invariant classes for renormalization

Let us fix some integer $d\geq 2$ and consider the class of functions

${\mathcal{F}}:=\left\\{f:Dom(f)\to\hat{\mathbb{C}}\;\middle|\;\begin{tabular}[]{@{}l@{}}$Dom(f)$
is an open subset of ${\mathbb{C}}$ containing\\\ $\\{0,\infty\\}$, $f$ is
analytic on $Dom(f)$, $f(0)=0,$\\\ $f(\infty)=\infty,$ and $f^{\prime}(0)=1.$
The restriction\\\ $f:f^{-1}(\mathbb{C}^{*})\to\mathbb{C}^{*}$ is a branched
covering\\\ map with a unique critical value $cv_{f}$, and\\\ all critical
points are of local degree $d$.\end{tabular}\right\\}.$

For example, $\mathcal{F}$ contains the unicritical polynomial

$G(z):=\frac{(z+1)^{d}-1}{d}.$

Fix some $f\in\mathcal{F},$ $\omega\in\mathcal{Q}$, and set
$q=\mathfrak{q}(\omega)$, $h=e^{2\pi i\mu_{\omega}(0)}f.$

######  Proposition 3.1.

$h$ has a non-degenerate $\omega$-parabolic fixed point at $z=0$.

###### Proof.

The uniqueness of the critical value of $h$ implies that the parabolic fixed
point is non-degenerate, see for example arguments in [Mil06] or [Shi00]. ∎

Let $P_{j}$ be petals for $h$ relative to $h$ with respective Fatou
coordinates $\phi_{j}$ as in the previous section. Denote by

$B=B^{h}=\bigcup_{m\geq 0}h^{-m}(P_{0})$

the parabolic basin of $h.$ Let $B_{0}=B^{h}_{0}$ be the immediate parabolic
basin of $h$, i.e. $B_{0}$ is the connected component of $B$ containing the
critical value $cv_{h}:=e^{2\pi i\mu_{\omega}(0)}cv_{f}$. After relabeling the
petals, while maintaining the cyclic ordering about $z=0$, we can assume
$P_{0}\subset B_{0}$. We can analytically extend the Fatou coordinate
$\phi_{0}$ to a map $\rho=\rho_{h}:B\to\mathbb{C}$ by defining

$\rho(z)=\phi_{0}(h^{mq+k}(z))-m$

for any integers $0\leq k<q$ and $m\geq 0$ such that $h^{mq+k}(z)\in P_{0}$.
For the rest of this paper, we will always choose the Fatou coordinate
$\phi_{0}$ so that $\rho(cv_{h})=0$. We can analytically extend the inverse of
the Fatou coordinate $\phi_{1}$ to a map $\chi=\chi_{h}$ by defining

$\chi(w)=h^{mq}((\phi_{1})^{-1}(w-m))$

for all sufficiently large integers $m\geq 0$. It follows from the definition
that the horn map $H=H_{h}^{h}$ has domain $\chi^{-1}(B)$ and satisfies

$H(w)=\rho\circ\chi(w).$

So defined, the horn map $H$ does not depend on the choice of petals and is
uniquely determined. Thus the parabolic renormalization $\mathcal{R}_{h}h$ is
canonically defined and has domain

$Dom(\mathcal{R}_{h}h):=\\{0,\infty\\}\cup\text{Exp}(\chi^{-1}(B)).$

We define the $(\omega,0)$-fiber renormalization of $f$ to be the map

$\mathcal{R}_{\omega,0}f:=\mathcal{R}_{h}h.$

The operator $\mathcal{R}_{\omega,0}$ is well-defined on $\mathcal{F}$, and
$\mathcal{F}$ is invariant under this operator:

######  Proposition 3.2.

For all $\omega\in\mathcal{Q}$,
$\mathcal{R}_{\omega,0}(\mathcal{F})\subset\mathcal{F}.$ Moreover, $z=1$ is
the unique critical value of $\mathcal{R}_{\omega,0}f$ for all
$f\in\mathcal{F}$.

###### Proof.

The proof in the case $\omega=\emptyset$ and $d=2$ is given in [Shi00], the
argument in the general case is identical. ∎

$H_{G}$
---
$\chi_{G}$
---
$\rho_{G}$
---
$\mathcal{R}_{\emptyset,0}G$
---
Exp
---
$\mathbb{C}$
---
$\mathbb{C}^{*}$
---
$B^{G}$
---
$\mathbb{C}$
---
Figure 3. The $\emptyset$-parabolic renormalization of $G$ when $d=2$.

We can similarly define the bottom fiber renormalization as
$\mathcal{R}_{\omega,0}^{\flat}f:=\mathcal{R}_{h}^{\flat}h$, the class
$\mathcal{F}$ is also invariant under this operator. All of the constructions
and statements in this section can be modified to replace top renormalizations
with bottom renormalizations; to simplify our presentation we will solely
focus on the top renormalizations.

We can more precisely describe the image of the fiber renormalization
operators $\mathcal{R}_{\omega,0}$. It is proved in [DH84] [DH85] that
$B^{G}$, the parabolic basin of the polynomial $G$, is a Jordan domain.
Consequently, $Dom(\mathcal{R}_{\emptyset,0}G)$ is the union of two open
Jordan domains, one containing $0$ and the other containing $\infty$ (see
Figure 3). Let $Dom_{0}(\mathcal{R}_{\emptyset,0}G)$ be the connected
component of $Dom(\mathcal{R}_{\emptyset,0}G)$ containing $0$, by the Riemann
mapping theorem there exists some $R_{G}>0$ and univalent map
$\varphi_{G}:\mathbb{D}_{R_{G}}\to Dom_{0}(\mathcal{R}_{\emptyset,0}G)$
satisfying $\varphi_{G}(0)=0$ and $\varphi_{G}^{\prime}(0)=1.$ For
$0\leq\delta<1$, let $\mathcal{S}_{\delta}$ be the set of univalent maps
$\varphi:\mathbb{D}_{(1-\delta)R_{G}}\to\mathbb{C}$ satisfying $\varphi(0)=0$
and $\varphi^{\prime}(0)=1$. We define the class of maps

$\mathcal{F}_{\delta}:=\\{\mathcal{R}_{\emptyset,0}G\circ\varphi_{G}\circ\varphi^{-1}:\varphi\in\mathcal{S}_{\delta}\\}.$

It follows from the Koebe distortion theorem that $\mathcal{S}_{\delta}$, and
consequently $\mathcal{F}_{\delta}$, is compact with respect to the compact-
open topology for all $0\leq\delta<1$.

By abuse of notation, for any classes $\mathcal{G},\mathcal{G}^{\prime}$ of
analytic maps we will write $\mathcal{G}\subset\mathcal{G}^{\prime}$ if every
map in $\mathcal{G}$ has a restriction in $\mathcal{G}^{\prime}$. In
particular, $\mathcal{F}_{\delta^{\prime}}\subset\mathcal{F}_{\delta}$ for any
$0\leq\delta^{\prime}\leq\delta$.

######  Proposition 3.3.

For all $\omega\in\mathcal{Q}$,
$\mathcal{R}_{\omega,0}(\mathcal{F})\subset\mathcal{F}_{0}\subset\mathcal{F}.$

###### Proof.

For $\omega=\emptyset$, this is proved in [Shi00] and [LY14]. The same
argument can be applied for general $\omega$. ∎

For any map $f$ which has a restriction in $\mathcal{F}_{0}$, we will denote
by $Dom_{0}(f)$ the domain on which $f$ restricts to a map in
$\mathcal{F}_{0}$.

While $\mathcal{F}_{\delta}$ is not contained in $\mathcal{F}$ for $\delta>0$,
if $\delta$ is sufficiently small then $z=1$ is the unique critical critical
value of every $f\in\mathcal{F}_{\delta}$. For any $\omega\in\mathcal{Q}$, it
follows from the compactness of $\mathcal{F}_{\delta}$ that if $\delta$ is
sufficiently small then for any $f\in\mathcal{F}$ and $h=e^{2\pi
i\mu_{\omega}(0)}f$ the orbit of the unique critical value of $h$ converges
asymptotically towards $z=0$. Similarly to the $f\in\mathcal{F}$ case, we can
canonically define $\mathcal{R}_{h}h$ as a function, so the $(\omega,0)$-fiber
renormalization operator

$f\mapsto\mathcal{R}_{\omega,0}f:=\mathcal{R}_{h}h$

is well-defined on $\mathcal{F}_{\delta}.$ When $\delta$ is sufficiently
small, the class $\mathcal{F}_{\delta}$ is also invariant under fiber
renormalization:

######  Theorem 3.4.

For all $\omega\in\mathcal{Q}$, if $\delta>0$ is sufficiently small then there
exists $0<\delta^{\prime}<\delta$ satisfying

$\mathcal{R}_{\omega,0}(\mathcal{F}_{\delta})\subset\mathcal{F}_{\delta^{\prime}}.$

###### Proof.

For $\omega=0$, this theorem is the main result in [Ché22]. We will not
recreate the full argument here; we will instead examine the two main steps
used by Chéritat and observe that the same reasoning applies to general the
$\omega$ case. For details, see [Ché22].

Let us fix some $f\in\mathcal{F}_{0}$, $\omega\in\mathcal{Q}$, and set
$h=e^{2\pi i\mu_{\omega}(0)}f$. The first part of the argument in [Ché22] is a
contraction: showing that for any $\delta>0$ sufficiently small there exists
some $0<\delta^{\prime}\ll\delta$ such that the restriction of
$\mathcal{R}_{\omega,0}f$ to $\mathcal{F}_{\delta^{\prime}}$ depends only on
the restriction of $f$ to $\mathcal{F}_{\delta}$. Proving this fact requires
showing that $B_{0}^{h}$ intersects only finitely many components of
$h^{-1}(\mathbb{C}^{*}\setminus\partial\mathbb{D})$, and comparing four
different metrics:

  1. (1)

The hyperbolic metric on $Dom_{0}(\mathcal{R}_{\omega,0}f)$.

  2. (2)

The hyperbolic metric on $B_{0}^{h}$.

  3. (3)

The box-Euclidean metric on $Dom_{0}(h)$, which is the pull-back of the flat
metric on $\mathbb{C}^{*}$ by $h$.

  4. (4)

The hyperbolic metric on $Dom_{0}(h)=Dom_{0}(f)$.

When we switch from $\omega=\emptyset$ to general $\omega\in\mathcal{Q}$, we
must consider the structure of $\bigcup_{n=0}^{q-1}B_{0}$ instead of just
$B_{0}^{h}$, where $q=\mathfrak{q}(\omega)$. In [Ché22], showing that
$B_{0}^{h}$ intersects only finitely many components of
$h^{-1}(\mathbb{C}^{*}\setminus\partial\mathbb{D})$ for $\omega=\emptyset$
follows from studying the geometry of the set
$\rho_{h}^{-1}(\mathbb{R}_{\geq-1})$. The same analysis can be applied to show
that $\bigcup_{n=0}^{q-1}B_{0}$ intersects only finitely many components of
$h^{-1}(\mathbb{C}^{*}\setminus\partial\mathbb{D})$ in the general $\omega$
case. The comparison of the four metrics is identical in the
$\omega=\emptyset$ and $\omega\neq\emptyset$ cases.

The second part of the argument in [Ché22] is a perturbation: showing that for
any $\delta$ sufficiently small, ${{f}}\in\mathcal{F}_{\delta}$, and
${h}:=e^{2\pi i\mu_{\omega}(0)}{f}$, we can be perturb $f$ to the restriction
of a map in $\mathcal{F}_{0}$ without moving the orbits of $h$ very far. The
argument here for $\omega\neq\emptyset$ is the sames as for the
$\omega=\emptyset$ case. ∎

The key feature of theorem 3.4 is that we can pick $\delta^{\prime}<\delta$.
As a consequence, the parabolic renormalization operators are contracting on
$\mathcal{F}$:

######  Corollary 3.5.

If
$\mathcal{R}_{\omega,0}(\mathcal{F}_{\delta})\subset\mathcal{F}_{\delta^{\prime}}$
for some $\omega\in\mathcal{Q}$ and $0<\delta^{\prime}<\delta$, then there is
a complete metric $\mathfrak{d}$ on $\mathcal{F}_{\delta^{\prime}}$ such that

$\mathfrak{d}(\mathcal{R}_{\omega,0}(f_{1}),\mathcal{R}_{\omega,0}(f_{2}))<\frac{1-\delta}{1-\delta^{\prime}}\mathfrak{d}(f_{1},f_{2})$

for all $f_{1},f_{2}\in\mathcal{F}_{\delta^{\prime}}$. Moreover, convergence
in this metric implies convergence in the compact-open topology.

###### Proof.

The inclusion $\mathcal{F}_{\delta^{\prime}}\subset\mathcal{F}_{\delta}$
induces an embedding of $\mathcal{F}_{\delta^{\prime}}$ into the the class of
maps

$\tilde{\mathcal{F}}_{\delta}:=\\{\mathcal{R}_{\emptyset,0}G\circ\varphi_{G}\circ\varphi^{-1}:\varphi\in\mathcal{S}_{\delta}\text{
and }\varphi\text{ has a quasiconformal extension to }\mathbb{C}\\}.$

The proof of Main Theorem 2 in [IS08] can applied to this setting, showing
that there is a one to one correspondence between
$\tilde{\mathcal{F}}_{\delta}$ and the Teichmüller space of
$\mathbb{C}\setminus\overline{\mathbb{D}_{(1-\delta)R_{G}}}$. Results in
[IS08] imply that the Teichmüller metric induces the desired metric
$\mathfrak{d}$ on
$\tilde{\mathcal{F}}_{\delta}\supset\mathcal{F}_{\delta^{\prime}}$. ∎

Let us now fix some $\omega\in\mathcal{Q}$, $0<\delta<\delta_{\omega}$,
$f\in\mathcal{F}_{\delta}$, and set $q=\mathfrak{q}(\omega)$,
$p=\mathfrak{p}(\omega)$, and $h_{\alpha}=e^{2\pi i\mu_{\omega}(\alpha)}f$ for
all $\alpha\in\mathbb{C}$. We also denote by $cv_{\alpha}=e^{2\pi
i\mu_{\omega}(\alpha)}$ the unique critical value of $h_{\alpha}$. We would
like to similarly define the $(\omega,\alpha)$-fiber renormalization of $f$ to
be

$\mathcal{R}_{\omega,0}f:=\mathcal{R}_{h_{0}}h_{\alpha}$

when $\alpha\in A_{1}$ is small, however we need to canonically define the
petals for $h_{\alpha}$ relative to $h_{0}$ and their respective Fatou
coordinates. To resolve this issue we make the following observation:

######  Proposition 3.6.

There exist constants $r,m_{0},M>0$, which all do not depend on $f$, such that
for any $\alpha\in A_{r}^{+}$ and $\theta\in[-\frac{\pi}{13},\frac{\pi}{13}]$
the petals $P_{h_{\alpha},j}^{h_{0}}$ with Fatou-tilt $\theta$ can be uniquely
defined so that:

  1. (1)

$m_{0}$ is the minimal integer such that
$h_{\alpha}^{m_{0}q+k}(cv_{\alpha})\in P_{h_{\alpha},2kp}^{h_{0}}$ for all
$0\leq k<q$.

  2. (2)

$h_{\alpha}^{(m_{0}-1)q+k}(cv_{\alpha})\in\partial P_{h_{\alpha},2kp}^{h_{0}}$
for all $0\leq k<q$.

  3. (3)

$M>0$ is the corresponding constant in proposition 2.4.

###### Proof.

For fixed $f$, we can find $r,m_{0},M>0$ which depend on $f$ so that the
proposition holds by proposition 2.4. The compactness of
$\mathcal{F}_{\delta}$ allows us to remove the dependence of these constants
on $f$. ∎

Proposition 3.6 gives canonical choices for the petals for $h_{\alpha}$
relative to $h_{0}$, up to specifying the Fatou-tilt. By setting

$\phi_{h_{\alpha},2kp}^{h_{0}}(h_{\alpha}^{m_{0}q+k}(cv_{\alpha}))=\begin{cases}m_{0}&\text{
if }k=0\\\ m_{0}+1&\text{ if }k\neq 0\end{cases}$

for all $0\leq k<q$, we can canonically choose the Fatou-coordinates. Note
that this choice agrees with the definition of $\rho_{h_{0}}$ in the
$\alpha=0$ case. With these choices, the near-parabolic renormalization
$\mathcal{R}_{h_{0}}h_{\alpha}$ is well-defined up to choosing the Fatou-tilt
of the petals. We define the $(\omega,\alpha)$-fiber renormalization of $h$ to
be the map

$\mathcal{R}_{\omega,\alpha}f:=\mathcal{R}_{h_{0}}h_{\alpha},$

defined on the maximal domain containing $z=0$ such that
$\mathcal{R}_{\omega,\alpha}f$ does not depend on the choice of Fatou-tilt
$\theta\in[-\frac{\pi}{13},\frac{\pi}{13}]$. Compactness of the closed
interval guarantees that $\mathcal{R}_{\omega,\alpha}f$ is defined on an open
neighborhood of $z=0$. We define the fiber renormalization in this way
specifically to that we are free to pick whatever Fatou-tilt we prefer, a
flexibility that may be useful in some applications. By similar construction
and argument in the $\alpha\in A_{1}^{-}$ case, there is an $r>0$ such that
the fiber renormalization operators $\mathcal{R}_{\omega,\alpha}$ are well-
defined on $\mathcal{F}_{\delta}$ for all $\alpha\in A_{r}$.

Perturbing theorem 3.4, we can produce classes invariant under these fiber
renormalizations:

######  Theorem 3.7.

For all $\omega\in\mathcal{Q}$, if $\delta>0$ is sufficiently then there exist
$r>0$ and $0<\delta^{\prime}<\delta$ such that

$\mathcal{R}_{\omega,\alpha}(\mathcal{F}_{\delta})\subset\mathcal{F}_{\delta^{\prime}}$

for all $\alpha\in A_{r}\cup\\{0\\}$.

###### Proof.

For $\alpha\in A_{r}^{+}$, an analogous result is proved in [IS08] for the
Inou-Shishikura class, we sketch the argument in our setting here. For details
see [Shi00].

Let us fix some $f\in\mathcal{F}_{\delta}$, $\omega\in\mathcal{Q}$, and set
$h_{\alpha}=e^{2\pi i\mu_{\omega}(\alpha)}f$ for all $\alpha\in
A_{1}^{+}\cup\\{0\\}$. Let us fix the Fatou-tilt of the petals for
$h_{\alpha}$ relative to $h_{0}$ to be $\theta=0$. For all $\epsilon>0$, set

$\Gamma_{\epsilon}:=\partial\mathbb{D}_{\epsilon}\cup\partial\mathbb{D}_{1/\epsilon}\cup\mathbb{R}_{>0}.$

The set $(\text{Exp}\circ\phi_{h_{\alpha},0}^{h_{0}})^{-1}(\Gamma_{\epsilon})$
cuts $P_{h_{\alpha},0}^{h_{0}}$ into connected components, which we will call
puzzle pieces. Taking inverse images under $h_{\alpha}$, we can find pre-
images of the puzzle pieces in $P_{h_{\alpha,1}}^{h_{0}}$, which we will call
pre-puzzle pieces, such that $\mathcal{R}_{\omega,\alpha}f$ is determined by
the induced map from pre-puzzle pieces to puzzle pieces. Moreover, if
$\epsilon>0$ is sufficiently small then the restriction of
$\mathcal{R}_{\omega,0}f$ to $\mathcal{F}_{\delta^{\prime}}$ is determined by
only finitely many pre-puzzle pieces.

The induced map from pre-puzzle pieces to
$\mathbb{C}^{*}\setminus\Gamma_{\epsilon}$ defines analytic isomorphisms from
the pre-puzzle pieces when $\alpha\neq 0$ to the pre-puzzle pieces when
$\alpha=0$. These isomorphisms induce an analytic map $\varphi$ such that
$(\mathcal{R}_{\omega,\alpha}f)\circ\varphi=\mathcal{R}_{\omega,0}f$ on the
restriction of $\mathcal{R}_{\omega,0}f$ to $\mathcal{F}_{\delta^{\prime}}$,
which shows that
$\mathcal{R}_{\omega,\alpha}f\in\mathcal{F}_{\delta^{\prime}}$ when $\alpha$
is sufficiently small.

By modifying $\Gamma_{\epsilon}$, we can repeat the argument for different
choices of Fatou-tilt. The compactness of $[-\frac{\pi}{13},\frac{\pi}{13}]$
allows us to remove the dependence on $\theta$. The compactness of
$\mathcal{F}_{\delta}$ allows us to remove the dependence on $f$.

The case $\alpha\in A_{r}^{-}$ can be handled similarly, and the ∎

As a consequence, we see that the fiber renormalizations are contracting in
the sense of corollary 3.5, with $\mathcal{R}_{\omega,0}$ replaced by
$\mathcal{R}_{\omega,\alpha}$. Additionally, we get a uniform version of
theorem 3.7:

######  Corollary 3.8.

For all $M\geq 0$ and sufficiently small $\delta>0$, there exists $r>0$ and
$0<\delta^{\prime}<\delta$ such that if $\omega\in\mathcal{Q}_{m}$ for some
$0\leq m\leq M$ and $\alpha\in A_{r}$, then

$\mathcal{R}_{\lfloor\omega\rfloor_{m^{\prime}},\alpha^{\prime}}(\mathcal{F}_{\delta})\subset\mathcal{F}_{\delta^{\prime}}$

for some $0\leq m^{\prime}\leq m$ and $\alpha^{\prime}\in A_{r}$ such that
$\mu_{\omega}(\alpha)=\mu_{\lfloor\omega\rfloor_{m^{\prime}}}(\alpha^{\prime})$.

###### Proof.

For $M=0$, corollary 3.8 follows immediately from theorem 3.7 as $\emptyset$
is the unique modified continued fraction of height $0$. Let us now fix some
$M=M_{0}>0$ and assume that corollary 3.8 holds for $M=M_{0}-1$. As
$\bigcup_{m=0}^{M}\mathcal{Q}_{m}$ is a finite union, the inductive hypothesis
allows us to restrict to the case $\omega\in\mathcal{Q}_{M}$.

Fix some $\omega=\langle
a_{j}:\varepsilon_{j}\rangle_{j=1}^{M}\in\mathcal{Q}_{M}$ and
$\alpha\in\mathbb{C}$. Proposition 1.3 implies that

$\displaystyle\mu_{\omega}(\alpha)$
$\displaystyle=\mu_{\lfloor\omega\rfloor_{m}}\left(\mathcal{E}(\lfloor\omega\rfloor_{m})\mu_{\lceil\omega\rceil^{m}}(\mathcal{E}(\omega)\mathcal{E}(\lceil\omega\rceil^{m})\alpha)\right)$
$\displaystyle=\mu_{\lfloor\omega\rfloor_{m}}\left(\frac{\varepsilon_{m+1}\mathcal{E}(\omega_{m})}{a_{m+1}+\mu_{\lceil\omega\rceil^{m+1}}(\mathcal{E}(\omega)\mathcal{E}(\lceil\omega\rceil^{m+1})\alpha)}\right)$
$\displaystyle:=\mu_{\lfloor\omega\rfloor_{m}}(\alpha_{m})$

for all $0\leq m<M$.

It follows from proposition 1.4 that for any $r>0$ there exists some $N>0$,
which does not depend on our choice of $\omega$ or $\alpha$, such that if
$\alpha\in A_{1}$ and if $a_{m+1}\geq N$ for any $0\leq m<M$, then
$\alpha_{m}\in A_{r}$. This case can therefore be handled by the inductive
hypothesis, so it remains to consider the case where $a_{m}\leq N$ for all
$0\leq m\leq M$ and some large $N\geq 0$. But this restricts $\omega$ to a
finite subset of $\mathcal{Q}_{M}$, so the we can pick $r,\delta$, and
$\delta^{\prime}$ uniformly. ∎

###  3.1. Near-parabolic renormalization

For all $\alpha\in A_{1/2}$ we define

$T(\alpha)=\begin{cases}-1/\alpha&\text{ if }\alpha\in A_{1/2}^{+}\\\
1/\alpha&\text{ if }\alpha\in A_{1/2}^{-}\end{cases}.$

Identifying the pair $(\alpha,f)$ with the map $e^{2\pi i\alpha}f$ for
$\alpha\in\mathbb{C}/\mathbb{Z}$ and $f\in\mathcal{F}_{\delta}$ for all
$0\leq\delta<1$, we define the $\omega$-near-parabolic renormalization to be
the skew product

$\mathcal{R}_{\omega}:(\mu_{\omega}(\alpha),f)\mapsto(T(\alpha),\mathcal{R}_{\omega,\alpha}f),$

taking the representative of $\mu_{\omega}(\alpha)$ in $\mathbb{C}/\mathbb{Z}$
so that $\alpha\in A_{1/2}$. In applications, this is the operator we are
usually most interested in as the dynamics of $(\alpha,f)$ and
$\mathcal{R}_{\omega}(\alpha,f)$ are related by proposition 2.11. As
$\mathcal{R}_{\omega,\alpha}$ is contracting for all $\alpha\in A_{r}$ and
$\mu_{\omega}(\alpha)\mapsto T(\alpha)$ is expanding, the near-parabolic
renormalization operator $\mathcal{R}_{\omega}$ is hyperbolic on
$\mu_{\omega}(A_{r})\times\mathcal{F}_{\delta}$ for $\delta,r>0$ sufficiently
small.

For all $N,M\geq 0$ and $\omega\in\mathcal{Q}_{\infty}$, we will say that the
irrational number $x=\mu_{\omega}(0)$ is $(N,M)$-valley-type if for every
$j\geq 1$ there exist some $j\leq j^{\prime}\leq M$ such that
$a_{j^{\prime}}>N$. We denote the set of all $(N,M)$-valley-type numbers by
$VT_{N,M}$. The $(N,0)$-valley-type numbers are often called high-type. The
valley-type numbers are invariant under the action of near-parabolic
renormalization on the first coordinate:

######  Proposition 3.9.

If $x=\mu_{\omega}(0)$ is $(N,M)$-valley-type for some
$\omega\in\mathbb{Q}_{\infty}$, then for any $m\geq 0$ there is an integer $k$
such that

$T\circ\mu_{\lfloor\omega\rfloor_{m}}^{-1}(x)+k\in VT_{N,M}.$

###### Proof.

Assume $\omega=\langle a_{j}:\varepsilon_{j}\rangle_{j=1}^{\infty}$. Setting

$\alpha:=\mathcal{E}(\lfloor\omega\rfloor_{m})\mu_{\lceil\omega\rceil^{m}}(0),$

it follows from proposition 1.3 that

$x=\mu_{\omega}(0)=\mu_{\lfloor\omega\rfloor_{m}}(\mathcal{E}(\lfloor\omega\rfloor_{m})\mu_{\lceil\omega\rceil^{m}}(0))=\mu_{\lfloor\omega\rfloor_{m}}(\alpha).$

It follows from the definition of $T$ that

$a_{m+1}+T(\alpha)=-\cfrac{\varepsilon_{m+2}}{a_{m+2}+\cfrac{\varepsilon_{m+3}}{\ddots}}\in
VT_{N,M}.$

∎

As a result of proposition 3.9, in the valley-type setting we can apply near-
parabolic renormalization operators infinitely often:

######  Proposition 3.10.

For any $M\geq 0$ and sufficiently small $\delta>0$, if $N\geq 0$ is
sufficiently large then for any $x\in VT_{N,M}$ there exists a sequence
$\\{\omega_{1},\omega_{2},\ldots\\}\subset\bigcup_{m=0}^{M}\mathcal{Q}_{m}$
such that

$\mathcal{R}_{\omega_{n}}\cdots\mathcal{R}_{\omega_{1}}(x,f)$

is defined for all $n\geq 0$ and $f\in\mathcal{F}_{\delta}$

###### Proof.

It follows from proposition 1.4 that for any $M\geq 0$ and $r>0$, there exists
some $N\geq 0$ so that for $x\in VT_{N,M}$, there exists
$\omega\in\bigcup_{m=0}^{M}\mathcal{Q}_{m}$ and $\alpha\in A_{r}$ such that
$x=\mu_{\omega}(\alpha)$. Corollary 3.8 and proposition 3.9 together imply
that there is some $\omega^{\prime}\in\bigcup_{m=0}^{M}\mathcal{Q}_{m}$ and
$x^{\prime}\in VT_{N,M}$ such that

$\mathcal{R}_{\omega^{\prime}}(\\{x\\}\times\mathcal{F}_{\delta})\subset\\{x^{\prime}\\}\times\mathcal{F}_{\delta}.$

We can therefore prove the proposition by induction. ∎

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Department of Mathematics, University of Michigan, East Hall, 530 Church
Street, Ann Arbor, Michigan 48109

E-mail address<EMAIL_ADDRESS>