2.1 Moduli space

Denote the moduli space of quadratic rational maps with a marked critical point by ${\mathcal {M}}_2^{\mathrm{cm}}$ . More precisely, consider dynamical pairs $(f\!,\omega )$ where ${f: \mathbb {P}^1_{\mathbb {C}}~\to \mathbb {P}^1_{\mathbb {C}}}$ is a quadratic rational map and $f'(\omega )=0$ . We say that two such pairs $(f\!,\omega )$ and $(g,\omega ')$ are conjugate if there exists a Möbius transformation $\gamma $ such that $\gamma \circ g = f \circ \gamma $ and $\gamma (\omega ') = \omega $ . According to Lemma 6.1 in [Reference MilnorMil93], the moduli space of critically marked quadratic rational maps ${\mathcal {M}}_2^{\mathrm{cm}}$ is an algebraic surface with a unique singular point at the conjugacy class of $(z \mapsto z^{-2},\omega =0)$ . We refer to the elements of ${\mathcal {M}}^{\mathrm{cm}}_2$ simply as maps. Also, ${\mathcal {S}}_p$ is an algebraic subvariety of ${\mathcal {M}}_2^{\mathrm{cm}}$ , for all $p\ge 1$ . The curves ${\mathcal {S}}_p$ are known to be smoothly embedded in ${\mathcal {M}}_2^{\mathrm{cm}}$ , with the sole exception of ${\mathcal {S}}_2$ at the singular point of ${\mathcal {M}}_2^{\mathrm{cm}}$ (see, for example, [Reference EpsteinEps, Reference Levin, Shen and van StrienLSvS19]; cf. [Reference Kiwi and ReesKR13, Theorem 2.5]). Conjecturally ${\mathcal {S}}_p$ is irreducible for all $p\ge 1$ (cf. [Reference RamadasRam24]).

2.2 Counting in the quadratic family

Abstractly, for $p \ge 1$ , the numbers $\eta '(p) \in \mathbb {N}$ are defined by

$$ \begin{align*}2^{p-1} = \sum_{d|p} \eta'(d).\end{align*} $$

Therefore,

$$ \begin{align*}\eta'(p) = \dfrac{1}{2} \sum_{d|p} \mu(d)\cdot 2^{p/d},\end{align*} $$

where $\mu $ is the Möbius function.

It is, however, convenient to relate $\eta '(p)$ with quadratic polynomial dynamics, as in the introduction. Recall that $Q_c(z) := z^2 +c$ denotes the quadratic family. Then

(2.1) $$ \begin{align} \eta'(p) = |\{ c \in \mathbb{C} : z=0 \text{ has period } p \text{ under } Q_c \}|. \end{align} $$

In fact, according to Gleason (see, for example, [Reference Douady and HubbardDH85, Exposé XIX]), all the roots of the degree $2^{p-1}$ polynomial $Q_c^{\circ p-1}(c)$ are simple. Thus, (2.1) holds.

In order to write a formula for $\eta ^{\prime }_{\mathrm{II}}(p)$ , we need to define $\nu _q(k)$ for $k \ge 1$ . Let r be such that $ r = k \mod q$ and $0 \le r < q$ . If $q \ge 2$ , then

$$ \begin{align*}\nu_q (k) := \begin{cases} \dfrac{1}{2} \cdot \dfrac{2^k-2^r}{2^q-1} & \text{ if } r \neq 0, \\ &\\ \dfrac{1}{2} + \dfrac{1}{2} \cdot \dfrac{2^k-1}{2^q-1}& \text{ if } r = 0. \end{cases} \end{align*} $$

For example,

(2.2) $$ \begin{align} \nu_2(k) := \begin{cases} \dfrac{2^k-2}6 & \text{ if } k \text{ is odd,}\\ \\ \dfrac{2^k +2}6 & \text{ if } k \text{ is even.} \end{cases} \end{align} $$

The numbers $\nu _q(k)$ are also closely related to the quadratic family. More precisely, $\nu _q(k)$ is the number of parameters c in the $1/q$ -limb of the Mandelbrot set for which the critical point $z = 0$ is periodic of period dividing k. Indeed, $2\nu _q(k)$ is exactly the number of angles $\theta \in [1/(2^q-1),\,2/(2^q-1)] \subset \mathbb {R}/\mathbb {Z}$ which are periodic and whose period divides k, under angle doubling. By the description of the Mandelbrot set due to Douady and Hubbard, this implies that $\nu _q(k)$ counts parameters c in the $1/q$ -limb for which $z=0$ is $Q_c$ -periodic with the corresponding periods (see [Reference Douady and HubbardDH85, Exposé XIV]; cf. [Reference MilnorMil00]).

Lemma 2.1. Let $\nu _2'(p)$ be such that

$$ \begin{align*}\nu_2 (p) = \sum_{2 \le d |p} \nu_2'(d).\end{align*} $$

Then $\eta '(p) = 3 \nu ^{\prime }_2(p)$ , for all $p\ge 3$ .

Proof. It is sufficient to prove that, for all $p \ge 3$ , we have

$$ \begin{align*}\sum_{3 \le d |p} \eta'(d) = 3 \sum_{3 \le d |p}{\nu^{\prime}_2(d)}.\end{align*} $$

Indeed, from (2.2), for p even,

$$ \begin{align*} \sum_{3 \le d |p} \eta'(d) = 2^{p-1} - \eta'(1) - \eta'(2) = 2^{p-1}-2 = 3 (\nu_2(p)-1) = 3 \sum_{3 \le d |p}{\nu^{\prime}_2(d)}.\end{align*} $$

A similar identity holds for p odd.

2.3 Counting in ${\mathcal {S}}_p$

A map $f \in {\mathcal {M}}_2^{\mathrm{cm}}$ is hyperbolic if both of its critical orbits are in the basin(s) of some attracting cycle(s). A connected component of hyperbolic maps is called a hyperbolic component. Following Rees [Reference ReesRee90], hyperbolic components are classified into four types according to the relative position of the critical points in the attracting basins. Here we are mainly concerned with type II components, also known as bitransitive or type B components. A type II hyperbolic component consists of maps having a unique attracting cycle of period $p \ge 2$ such that both critical points belong to periodic Fatou components. These Fatou components are necessarily distinct. Each type II hyperbolic component ${\mathcal {H}}$ of period p contains a unique postcritically finite map $f_0$ called the center of ${\mathcal {H}}$ . The critical points of $f_0$ lie in the same orbit of period p. In particular, $f_0 \in {\mathcal {S}}_p$ .

In [Reference Kiwi and ReesKR13] we obtained a formula for the number $\eta _{\mathrm{II}}'(p)$ of type II hyperbolic components in ${\mathcal {M}}_2^{\mathrm{cm}}$ of exact period p. It is not difficult to check that the formula written below is equivalent to the one obtained in [Reference Kiwi and ReesKR13, Theorem 1.2].

Theorem 2.2. For $p \ge 3$ , let $\delta (p) = 1$ if p is even, and $\delta (p)=0$ otherwise. Then

$$ \begin{align*} \sum_{3 \le d|p} \dfrac{\eta^{\prime}_{\mathrm{II}} (d)}{d}= \dfrac{\nu_2(p)}{2}-\dfrac{\delta(p)}{2} + \dfrac{A(p)}{2p},\end{align*} $$

where

$$ \begin{align*}A(p) := \sum_{q=2}^{p-1} \phi(q) \bigg( \sum_{k=1}^{p-1} \nu_q(k) (2^{p-k-1}- \nu_q(p-k))\bigg)\end{align*} $$

and $\phi $ denotes the Euler totient function.

The Möbius inversion formula yields

$$ \begin{align*}\eta^{\prime}_{\mathrm{II}} (p) = \dfrac{1}{2} \bigg( p \nu^{\prime}_2(p) + \sum_{d|p, p/d \ge 3} d \cdot \mu(d) \cdot A \bigg(\dfrac{p}{d}\bigg) \bigg).\end{align*} $$

2.4 Partial parametrization of ${\mathcal {M}}_2^{\mathrm{cm}}$

The complement of ${\mathcal {S}}_1 \cup {\mathcal {S}}_2$ in ${\mathcal {M}}_2^{\mathrm{cm}}$ is identified with $\mathbb {C}^* \times \mathbb {C}$ via the family

$$ \begin{align*}f_{a,b}(z) = 1 + \dfrac{b}{z} + \dfrac{a}{z^2}\end{align*} $$

with marked critical point $\omega =0$ . That is,

$$ \begin{align*}{\mathcal{M}}_2^{\mathrm{cm}} \setminus ({\mathcal{S}}_1 \cup {\mathcal{S}}_2) = \{ [(f_{a,b},0)] : (a,b) \in \mathbb{C}^* \times \mathbb{C}\}.\end{align*} $$

In fact, observe that if $[(f,\omega )] \in {\mathcal {M}}_2^{\mathrm{cm}} \setminus ({\mathcal {S}}_1 \cup {\mathcal {S}}_2)$ , then $\omega , f(\omega )$ and $f^{\circ 2}(\omega )$ are distinct. Normalizing so that $\omega =0$ , $f(\omega )=\infty $ and $f^{\circ 2}(\omega )=1$ , we have that $f=f_{a,b}$ , for some $(a,b) \in \mathbb {C}^* \times \mathbb {C}$ .

Regard the projective plane $\mathbb {P}^2_{\mathbb {C}}$ as a compactification of ${\mathcal {M}}_2^{\mathrm{cm}} \setminus ({\mathcal {S}}_1 \cup {\mathcal {S}}_2)$ via the inclusion $\mathbb {C}^* \times \mathbb {C} \hookrightarrow \mathbb {P}^2_{\mathbb {C}}$ given by $(a,b) \mapsto [a:b:1]$ . Thus, ${\mathcal {M}}_2^{\mathrm{cm}} \setminus ({\mathcal {S}}_1 \cup {\mathcal {S}}_2)$ is identified with the complement, in $\mathbb {P}^2_{\mathbb {C}}$ , of the lines

$$ \begin{align*}{\mathcal{L}}_+ := \{ [a:b:c] : a=0\}\end{align*} $$

and

$$ \begin{align*}{\mathcal{L}}_- := \{ [a:b:c]: c = 0 \}.\end{align*} $$

From now on we regard ${\mathcal {S}}_p$ , for any $p \ge 3$ , as a subset of $\mathbb {P}^2_{\mathbb {C}}$ and denote its closure by $\overline {{\mathcal {S}}}_p$ .

Milnor [Reference MilnorMil93] introduced a compactification of the moduli space of quadratic rational maps, without critical markings, by a line at infinity corresponding to pairs $\{\unicode{x3bb} ,\unicode{x3bb} ^{-1}\}$ where $\unicode{x3bb} \in \overline {\mathbb {C}}$ . Points in moduli space close to $\{\unicode{x3bb} ,\unicode{x3bb} ^{-1}\}$ have two fixed points with multipliers close to $\unicode{x3bb} $ and $\unicode{x3bb} ^{-1}$ . Silverman [Reference SilvermanSil98] showed that, in fact, Milnor’s compactification is the geometric invariant theory compactification of unmarked moduli space. Maps close to ${\mathcal {L}}_-$ and ${\mathcal {L}}_+$ in marked moduli space, after forgetting the marking, become maps in unmarked moduli space close to infinity. The line ${\mathcal {L}}_-$ should be regarded as a blow-up of $\{-1,-1\}$ . More precisely, as $[a:b:1]$ approaches $[1:m:0] \in {\mathcal {L}}_-$ with $m \neq 0$ , the multipliers of two fixed points of $f_{a,b}$ converge to $-1$ . Also, as $[a:b:1]$ approaches $[0:m:1] \in {\mathcal {L}}_+$ with $m\neq 0$ , the multipliers of two fixed points of $f_{a,b}$ converge to reciprocal complex numbers in $\mathbb {C} \setminus ~\{0,-1\}$ .

2.5 Degree of ${\mathcal {S}}_p$

Lemma 2.3. [Reference Kiwi and ReesKR13, Lemma 3.1]

For all $p \ge 3$ ,

$$ \begin{align*}\deg \overline{{\mathcal{S}}}_p = \nu^{\prime}_2(p).\end{align*} $$

Moreover, none of the points $[1:0:0],[0:1:0],[0:0:1]$ belong to $\overline {{\mathcal {S}}}_p$ .

Lemma 2.1 yields

(2.3) $$ \begin{align} \deg \overline{{\mathcal{S}}}_p = \dfrac{\eta'(p)}{3}. \end{align} $$

We remark that [Reference Kiwi and ReesKR13, Lemma 3.1] states an equivalent assertion but for $\overline {{\mathcal {X}}_n}:=\bigcup _{3 \le p|n} \overline {{\mathcal {S}}_p}$ instead of $\overline {{\mathcal {S}}}_p$ .

2.6 Punctures

Recall that the smooth compactification $\widehat {{\mathcal {S}}_p}$ of ${\mathcal {S}}_p$ is obtained from ${\mathcal {S}}_p$ by filling each of its punctures with an ideal point. Equivalently, it is the normalization of the projective curve $\overline {{\mathcal {S}}}_p$ . The punctures of ${\mathcal {S}}_p$ correspond to germs of branches of $\overline {{\mathcal {S}}}_p$ at the lines ${\mathcal {L}}_+$ and ${\mathcal {L}}_-$ . A representative of the branch corresponding to a puncture x will be denoted by ${\mathcal {E}}_x$ . A punctured neighborhood of x in $\widehat {{\mathcal {S}}_p}$ is ${\mathcal {E}}^*_x := {\mathcal {E}}_x \setminus {\mathcal {L}}_\pm $ .

Let $\pi : \widehat {{\mathcal {S}}_p} \to \overline {{\mathcal {S}}}_p$ be the natural projection. It is a bijection in ${\mathcal {S}}_p$ and maps each puncture ${x}$ onto the intersection $\pi (x)$ of the corresponding branch with ${\mathcal {L}}_+$ or ${\mathcal {L}}_-$ . We will systematically abuse notation and simply say that $x \in {\mathcal {L}}_+$ or $x \in {\mathcal {L}}_-$ .

We define the multiplicity $\mu _x$ of a puncture x as the intersection multiplicity of the branch ${\mathcal {E}}_x$ corresponding to x with the lines ${\mathcal {L}}_+, {\mathcal {L}}_-$ .

As a direct consequence of Bézout’s theorem and Lemma 2.3 we have the following result.

Lemma 2.4. For all $p \ge 3$ ,

$$ \begin{align*}\dfrac{\eta'(p)}{3} = \sum_{x \in {\mathcal{L}}_+} \mu_x = \sum_{x \in {\mathcal{L}}_-} \mu_x,\end{align*} $$

where the sums are taken over $ x \in \widehat {{\mathcal {S}}_p} \setminus {\mathcal {S}}_p$ .

3.1 Statement of the main lemma

For $(a,b) \in \mathbb {C}^* \times \mathbb {C}$ , and for all $n \ge 0$ , let

$$ \begin{align*}\omega_n (a,b) : = f^{\circ n}_{a,b} (0).\end{align*} $$

Partial derivatives will be denoted by $\partial _a \omega _n$ and $\partial _b \omega _n$ . In dynamical space, we are concerned with the behavior of

$$ \begin{align*} \dfrac{d f^{\circ p-1}_{a,b}}{dz} (\infty),\end{align*} $$

under the agreement that the above quantity is the derivative of $f^{\circ p-1}_{a,b} (1/z)$ at $z=0$ . Then $\partial _b \omega _p$ and $ ({d f^{\circ p-1}_{a,b}}/{dz}) (\infty )$ are meromorphic (rational) functions from $\widehat {{\mathcal {S}}}_p$ to $\overline {\mathbb {C}}$ . Given a meromorphic function $h: \widehat {{\mathcal {S}}}_p \to \overline {\mathbb {C}}$ , we denote by $\mathrm {ord}_y\ h \in \mathbb {Z}$ its order at $y \in \widehat {{\mathcal {S}}}_p$ . The order at a zero of h is positive and at a pole is negative.

The key to computing the Euler characteristic is to understand the asymptotics of $\partial _b \omega _p$ at the punctures.

Lemma 3.1. (Main)

Let $x \in \widehat {{\mathcal {S}}_p}$ be a puncture of ${\mathcal {S}}_p$ . Then

(3.1) $$ \begin{align} \mathrm{ord}_x\ \partial_b \omega_p = \mathrm{ord}_x\ \dfrac{d f^{\circ p-1}_{a,b}}{dz} (\infty) + \begin{cases} 0 & \text{ if } x \in {\mathcal{L}}_+, \\ 2 \mu_x & \text{ if } x \in {\mathcal{L}}_-.\\ \end{cases} \end{align} $$

The lemma is proved in §5.

3.2 Meromorphic $1$ -form and Euler characteristic

The curve ${\mathcal {S}}_p$ is contained in the vanishing locus of $\omega _p(a,b)$ . The gradient of $\omega _p$ is non-vanishing in ${\mathcal {S}}_p$ (see [Reference EpsteinEps, Reference Levin, Shen and van StrienLSvS19]). Thus, the differential of $\omega _p$ yields a meromorphic $1$ -form in $\widehat {{\mathcal {S}}_p}$ as follows.

Proposition 3.2. For all $(a,b) \in {\mathcal {S}}_p$ ,

$$ \begin{align*}(\partial_a \omega_p (a,b), \partial_b \omega_p (a,b)) \neq (0,0).\end{align*} $$

Moreover,

$$ \begin{align*} d \tau := \begin{cases} \dfrac{da}{\partial_b \omega_p (a,b)} & \text{ if } \partial_b \omega_p (a,b) \neq 0, \\ \\ - \dfrac{d b}{\partial_a \omega_p (a,b)} & \text{ if } \partial_a \omega_p (a,b) \neq 0, \end{cases}\end{align*} $$

is a well-defined non-vanishing holomorphic $1$ -form in ${\mathcal {S}}_p$ which extends to a meromorphic $1$ -form in $\widehat {{\mathcal {S}}_p}$ .

We omit the proof of the proposition which is a direct consequence of the transversality results cited above.

Proof of Theorem A, assuming Lemma 3.1

We prove the formula for the Euler characteristic. To simplify notation let

$$ \begin{align*}G (a,b) := \dfrac{d f^{\circ p-1}_{a,b}}{dz}(\infty).\end{align*} $$

The number of zeros and poles of the rational map $G:\widehat {{\mathcal {S}}}_p \to \overline {\mathbb {C}}$ agree, taking into account multiplicities. By the chain rule, the zeros of G in ${\mathcal {S}}_p$ are located at the parameters where the free critical point is in the orbit of $\omega =0$ . That is, the zeros of G are the centers of type II components of period p. We postpone proving that all the zeros of G are simple to the lemma below. Since the remaining poles and zeros of G are at the punctures, we have

$$ \begin{align*}\sum_{x \in \widehat{{\mathcal{S}}}_p \setminus {\mathcal{S}}_p} \mathrm{ord}_x\ G + \eta_{\mathrm{II}}'(p) = 0.\end{align*} $$

Now we compute the total degree of the $1$ -form $d \tau $ . Let x be a puncture of multiplicity $\mu _x$ . If $x \in {\mathcal {L}}_\pm $ , then there exists a local uniformizing parameter s such that $s^{\pm \mu _x} =a$ (cf. §4.3). Therefore,

$$ \begin{align*}da = \pm \mu_x {s^{\pm \mu_x-1}}ds.\end{align*} $$

Hence,

$$ \begin{align*}\mathrm{ord}_x\ d\tau = -1 \pm \mu_x - \mathrm{ord}_x\ \partial_b \omega_p,\end{align*} $$

In view of Lemma 3.1,

$$ \begin{align*} \sum_{x \in {\mathcal{L}}_+} \mathrm{ord}_x\ d\tau & = -\sum_{x \in {\mathcal{L}}_+} 1 + \sum_{x \in {\mathcal{L}}_+} \mu_x - \sum_{x \in {\mathcal{L}}_+} \mathrm{ord}_x\ G, \\ \sum_{x \in {\mathcal{L}}_-} \mathrm{ord}_x\ d\tau & = -\sum_{x \in {\mathcal{L}}_-} 1 - \sum_{x \in {\mathcal{L}}_-} \mu_x - \sum_{x \in {\mathcal{L}}_-} \mathrm{ord}_x\ G - 2 \sum_{x \in {\mathcal{L}}_-} \mu_x. \end{align*} $$

From Lemma 2.4, the sum of the multiplicities of the punctures in each line is $\eta '(p)/3$ and we obtain:

$$ \begin{align*}-\chi(\widehat{{\mathcal{S}}_p})= \sum_x \mathrm{ord}_x\ d \tau = -N_p -\dfrac{2 \eta'(p)}{3}+ \eta_{\mathrm{II}}'(p).\end{align*} $$

The desired formula for $\chi ({\mathcal {S}}_p)$ follows.

Proof. Let $\nu =\nu (a,b):=-2a/b$ be the free critical point of $f_{a,b}$ and $\nu _j := f^{\circ j}_{a,b} (\nu ).$ Given a zero $c=(a,b)$ of G, there exists $1 \le j < p$ such that $\omega _j(a,b) = \nu (a,b)$ . Close to c, there exists a non-vanishing holomorphic function $u(a,b)$ such that

$$ \begin{align*}G(a,b)=u(a,b) \cdot f^{\prime}_{a,b}(\omega_j(a,b)).\end{align*} $$

Since $\nu (c)$ is a simple critical point of $f_c$ , the multiplicities of $c \in {\mathcal {S}}_p$ as a zero of $\omega _j - \nu :{\mathcal {S}}_p \to \mathbb {C}$ and as a zero of G coincide. According to [Reference Levin, Shen and van StrienLSvS19], the intersection at c of the curves $\omega _j-\nu =0$ and ${\mathcal {S}}_p$ (i.e., $\omega _p-\omega _0=0$ ) is transversal. Hence, $c \in {\mathcal {S}}_p$ is a simple zero of $\omega _j-\nu :{\mathcal {S}}_p \to \mathbb {C}$ , and therefore it is a simple zero of G.

4.1 Fixed point multipliers

Generically a quadratic rational map has three fixed points. According to Epstein [Reference EpsteinEps00, Proposition 1] (cf. [Reference MilnorMil93, Lemma 4.1]), as $f \in {\mathcal {S}}_p$ converges to a puncture, one of the fixed point multipliers converges to $\infty $ , while the other two converge to conjugate roots of unity of order $q \ge 2$ . Observe that as $(a,b) \equiv [a:b:1]$ converges to $[1:\gamma : 0]\in {\mathcal {L}}_-$ , we have convergence of $a^{-1/2}f_{a,b}(a^{1/2} z)$ to $1/z$ , uniformly in $\mathbb {C}^*$ . Therefore, $f_{a,b}$ has a pair of fixed point multipliers converging to $-1$ . As $(a,b) \to (0,b) \neq (0,0)$ , we have convergence of $f_{a,b}$ to $1 +b/z$ , uniformly in $\mathbb {C}^*$ . Thus, the fixed point multipliers of f converge to $\infty $ and to the fixed point multipliers of $1+b/z$ . If the map $z \mapsto 1+b/z$ has finite order, then its order is at least $3$ . A straightforward calculation, (see, for example, [Reference Kiwi and ReesKR13, Lemma 3.4]) yields the following lemma.

Lemma 4.2. Consider $\theta \in \mathbb {Q} / \mathbb {Z}$ of order $q \ge 3$ . The fixed point multipliers of ${z \mapsto 1 +b/z}$ are $\exp (\pm 2 \pi i \theta )$ if and only if

$$ \begin{align*}b =-(2 \cos \pi \theta)^{-2}.\end{align*} $$

For $\theta \in \mathbb {Q}/\mathbb {Z}$ of order at least $3$ , let

$$ \begin{align*}c_\theta := -(2 \cos \pi \theta)^{-2}.\end{align*} $$

Every point in $\overline {{\mathcal {S}}}_p \cap {\mathcal {L}}_+$ is of the form $[0:c_\theta :1]$ for some $\theta \in \mathbb {Q} / \mathbb {Z}$ of order at least $3$ .

4.2 Parabolic parameter rescalings

Loosely, the term ‘parameter rescalings’ refers to families of maps that arise in suitable partial compactifications of moduli space. For example, De Marco [Reference DeMarcoDeM07] has shown that the resolution of moduli space iterate maps (for quadratic rational maps) is achieved exactly in the compactification of moduli space by all parabolic rescaling limits. A nice illustration of this phenomenon comes from a straightforward computation which we leave to the reader.

Proposition 4.3. Assume that $(a,b) \equiv [a:b:1]$ converges to $[1:\gamma : 0]\in {\mathcal {L}}_-$ with $\gamma \neq ~0$ . Then

$$ \begin{align*}\gamma \cdot f^{\circ 2}_{a,b} ({z}/{\gamma}) \to R_{\gamma}\end{align*} $$

uniformly in compact subsets of $\mathbb {C} \setminus \{-1\}$ .

That is, ${\mathcal {L}}_-$ is identified with $\mathrm {Per}_1^{\mathrm{cm}}(1) \setminus {\mathcal {S}}_1$ via a period $2$ rescaling limit.

The corresponding result for ${\mathcal {L}}_+$ can be roughly described as follows. Let $b_0 \in \mathbb {C}$ be such that $z \mapsto 1+b_0/z$ has order $q \ge 3$ . If we blow up $\mathbb {P}^2_{\mathbb {C}}$ once at $[0:b_0:1]$ , then the exceptional divisor is naturally identified with $\mathrm {Per}_1^{\mathrm{cm}}(1)$ via a period $q$ rescaling limit.

Proposition 4.4. Consider $\theta \in \mathbb {Q} / \mathbb {Z}$ of order $q \ge 3$ . Let

$$ \begin{align*} \beta_\gamma (a) &= c_\theta + \gamma a + o(a) \quad \text{ as } a \to 0, \\ M_a (z) &= a^{-1} c_\theta z. \end{align*} $$

Then there exists an invertible affine function $v(\gamma )=A\gamma +B$ such that, for all $\gamma \in \mathbb {C}$ , as $a \to 0$ ,

$$ \begin{align*}M_a \circ f^{\circ q}_{a,\beta_\gamma (a)} \circ M^{-1}_a \to R_{v(\gamma)}\end{align*} $$

uniformly in compact subsets of $\mathbb {C} \setminus \{-1\}$ .

This proposition is a result by Stimson, but in different coordinates. The required change of coordinates is discussed in [Reference Kiwi and ReesKR13]. We deduce the proposition from the results in [Reference Kiwi and ReesKR13]. Equivalent discussions are included in [Reference DeMarcoDeM07, Reference EpsteinEps00].

Proof. Consider the family of (totally marked) quadratic rational maps $h_{\rho ,\zeta }$ , defined for $(\zeta ,\rho ) \in \mathbb {C}^* \times \mathbb {C} \setminus \{0,-1,-2\}$ by

$$ \begin{align*} h_{\zeta\!, \rho}(z) :=& \ \zeta z\bigg(1-\dfrac{2+\rho}{2(1+\rho)} z\bigg)\bigg(1-\dfrac{2}{2+\rho} z\bigg)^{-1} \\=& \ \zeta z\bigg(1-\dfrac{\rho^2 z}{4(1+\rho)(1+\frac{1}{2} \rho-z)}\bigg). \end{align*} $$

Here, critical points and fixed points are marked. Indeed, $h_{\zeta \!,\rho }$ has a fixed point at $z=0$ with multiplier $\zeta $ , another fixed point at $z=\infty $ , a critical point at $1$ , and another critical point at $1+\rho $ . For $\rho $ small, the third fixed point is close to $z=1+\rho /2$ . Note that $h_{\zeta \!,\rho } \to ~\zeta z$ , as $\rho \to 0$ , uniformly in compact subsets of $\overline {\mathbb {C}} \setminus \{1\}$ .

There exists a sufficiently small a neighborhood $U \subset \mathbb {C}^2$ of $(\exp (2\pi i \theta ),0)$ such that, for all $(\zeta ,\rho ) \in U$ with $\rho \neq 0$ , we have that $1, h_{\zeta \!,\rho }(1)$ and $h^{\circ 2}_{\zeta \!,\rho }(1)$ are pairwise distinct. Thus, for all $(\zeta ,\rho ) \in U$ with $\rho \neq 0$ , there exist unique $a=a(\zeta ,\rho )$ and ${ b=b(\zeta ,\rho )}$ such that $h_{\zeta \!,\rho }$ is conjugate to $f_{a,b}$ , via the Möbius conjugacy mapping $1$ , $h_{\zeta \!,\rho }(1)$ and $h^{\circ 2}_{\zeta \!,\rho }(1)$ to $0$ , $\infty $ and $1$ , respectively. It is not difficult to conclude that the map ${(\zeta ,\rho ) \mapsto (a(\zeta ,\rho ),b(\zeta ,\rho ))}$ extends to a biholomorphic map H from U onto a neighborhood $U'$ of $(0,c_\theta )$ (see, for example, the proof of Lemma 3.7 in [Reference Kiwi and ReesKR13]). Under H, the line $\rho =0$ maps into $a=0$ .

According to Stimson (Theorem 3.6 in [Reference Kiwi and ReesKR13]), for any $m \in \mathbb {C}$ , if

$$ \begin{align*}\zeta = \exp(2 \pi i \theta) (1 + m \rho) + o(\rho)\end{align*} $$

then

$$ \begin{align*}\dfrac{h^{\circ q}_{\zeta\!, \rho} (1+z\rho)-1}{\rho} \to qm + z + \dfrac{1}{4(z-1/2)}\end{align*} $$

uniformly in compact subsets of $\mathbb {C} \setminus \{1/2\}$ . Note that the limit is conjugate to $R_v(w)$ with $v=-2 q m + 1$ via the change of coordinates $z=-w/2$ . Using the derivative of the biholomorphic map H, given $\gamma $ , there exists an invertible map $m(\gamma ):=\alpha \gamma +\beta $ such that if

$$ \begin{align*}b=c_\theta + \gamma a +o(a)\end{align*} $$

then

$$ \begin{align*}\zeta(a,b) = \exp(2 \pi i \theta) (1 + m(\gamma) \rho(a,b)) + o(\rho(a,b)),\end{align*} $$

as $a \to 0$ . Hence, there exists $M_{a,b}$ such that

$$ \begin{align*}M_{a,b} \circ f_{a,b}^{\circ q} \circ M_{a,b}^{-1} \to R_{v(\gamma)}\end{align*} $$

as $a \to 0$ , where

$$ \begin{align*}v(\gamma):=-2 q m(\gamma) +1.\end{align*} $$

Moreover, note that $M_{a,b}$ can be chosen such that $M_{a,b}(0)= 0$ , $M_{a,b} (-2a/c_\theta ) = -2$ and $M_{a,b}(\infty ) = \infty $ . Thus, we may choose $M_{a,b} (z)=a^{-1}c_\theta z=M_a(z)$ .

4.3 Local parametrization

To compute the order of various quantities at a puncture x it is convenient to work with a local parametrization of ${\mathcal {E}}_x$ .

Denote by $\pi _a: \mathbb {C}^* \times \mathbb {C} \to \mathbb {C}$ the projection onto the first coordinate. Then the multiplicity of x, denoted by $\mu _x$ , is the degree of the covering $\pi _a : {\mathcal {E}}_x^* \to D^*$ where $D^*$ is a sufficiently small punctured disk neighborhood of $0$ or $\infty $ and ${\mathcal {E}}_x$ is a suitable representative of the branch corresponding to x. Note that $D^*$ is a neighborhood of $0$ if $x \in {\mathcal {L}}_+$ , and of $\infty $ if $x \in {\mathcal {L}}_-$ . Since all degree $\mu _x$ coverings of a punctured disk are equivalent, it follows that there exist $\varepsilon>0$ and a power series

$$ \begin{align*}b_x(s):=\sum_{j \ge j_0} c_j s^{j}\!,\end{align*} $$

convergent in $\mathbb {D}_\varepsilon ^*$ such that

$$ \begin{align*}\begin{array}{ccc} \mathbb{D}_\varepsilon^*&\to &{\mathcal{E}}_x^*\\ s &\mapsto & (s^{\pm \mu_x}, b_x(s)) \end{array} \end{align*} $$

is a conformal isomorphism, for a choice of ${\mathcal {E}}_x$ . The series $b_x(s)$ is unique up to a $\mu _x$ -root of unity $\eta $ . That is, up to replacement of $c_j$ by $\eta ^{\pm j} c_j$ , for all j. We say that $(s^{\pm \mu _x}\!, b_x(s))$ is a local parametrization of ${\mathcal {S}}_p$ at x, and sometimes we will just say that $b_x$ is a local parametrization.

For future reference, we record a direct consequence of Lemmas 2.3 and 4.2.

Lemma 4.5. If $(s^{-\mu },b(s))$ is a local parametrization of a branch at ${\mathcal {L}}_-$ , then, as $s \to 0$ ,

$$ \begin{align*}b(s) = \dfrac{c}{s^{\mu}} + o(s^{-\mu}).\end{align*} $$

If $(s^{\mu },b(s))$ is a local parametrization of a branch at ${\mathcal {L}}_+$ , then, as $s \to 0$ ,

$$ \begin{align*}b(s) = c_\theta + c {s^{\mu}} + o(s^{\mu}).\end{align*} $$

4.4 Complex transversality

The main lemma is, in a certain sense, a manifestation of the following transversality results for the families $\{R_v\}$ and $\{Q_c\}$ .

Lemma 4.6. Consider $c_0 \in \mathbb {C}$ and $q_0 \ge 1$ such that $Q_{c_0}^{\circ q_0} (0) =0$ and $q_0$ is the smallest positive integer with this property. Then

$$ \begin{align*}\dfrac{d Q^{\circ q_0}_{c}(0)}{dc} (c_0) \neq 0.\end{align*} $$

Consider $v_0 \in \mathbb {C}$ and $\ell _0 \ge 1$ such that $R_{v_0}^{\circ \ell _0} (0) = 0$ or $-1$ and $\ell _0$ is the smallest positive integer with this property. Then

$$ \begin{align*}\dfrac{d R^{\circ \ell_0}_{v}(0)}{dv} (v_0) \neq 0.\end{align*} $$

5.1 From Puiseux series to complex numbers

Let us first prove that parameters ${\beta \in \mathbb {C} \langle \langle t \rangle \rangle }$ such that the critical point $\unicode{x3c9} $ is periodic under $\varphi _\beta $ are in correspondence with parametrizations of branches of periodic curves at ${\mathcal {L}}_\pm $ . The proof is essentially an immediate consequence of the fact that formal solutions $\beta \in \mathbb {C} \langle \langle t \rangle \rangle $ to the equation $f_{t^{\pm 1},b}^{\circ p}(0)=0$ , in the unknown b, are automatically ‘convergent’ (see, for example, [Reference FischerFis01, §7.2]).

Proof of Lemma 5.1

It is sufficient to show that $\beta _0$ is the Puiseux series of a branch of a periodic curve with period dividing p if and only if $\unicode{x3c9} =0$ is periodic of period dividing p under $\varphi _{\beta _0}$ .

Assume that $(s^{\pm \mu }, b(s))$ is a local parametrization of ${\mathcal {E}}_x$ for some puncture x at ${\mathcal {L}}_\pm $ of a periodic curve of period dividing p. Then $f_{s^{\pm \mu }, b(s)}^{\circ p} (0) =0,$ where

$$ \begin{align*}f_{s^{\pm \mu}, b(s)} (z) = 1 + \dfrac{b(s)}{z} + \dfrac{s^{\pm \mu}}{z^2}\in \mathbb{C}((s))(z).\end{align*} $$

Via the isomorphism $\bar {\iota }: \mathbb {C}((s))(z) \to \mathbb {C}((t^{1/\mu }))(z)$ that sends s to $t^{1/\mu }$ , we have that $f_{s^{\pm \mu }, b(s)}(z)$ becomes $\varphi _{\beta _0}(z)$ , where $\beta _0 = b(t^{1/\mu })$ . Thus, $\varphi _{\beta _0}^{\circ p} (0) =0$ . Hence, $\unicode{x3c9} =0$ is periodic with period dividing p, under iterations of $\varphi _{\beta _0}$ .

Conversely, assume that $\unicode{x3c9} =0$ has period dividing p, under $\varphi _{\beta _0}$ , for some $\beta _0\in \mathbb {C} \langle \langle t \rangle \rangle $ . Let $\mu \ge 1$ be minimal such that

$$ \begin{align*}\beta_0 = \sum_{j \ge j_0} c_j t^{j/\mu} \in \mathbb{C}((t^{1/\mu})).\end{align*} $$

Via $\bar {\iota }^{-1}$ , we have that $\varphi _{\beta _0}(z)$ becomes $f_{s^{\pm \mu },b_0(s)}(z)$ where

$$ \begin{align*}b_0(s) = \sum_{j \ge j_0} c_j s^j \in \mathbb{C}((s)).\end{align*} $$

Hence, $(s^{\pm \mu }, b_0(s))$ is a formal solution of $f_{s^{\pm \mu },b(s)}^{\circ p}(0) =0$ . Every formal solution is convergent, for $|s|$ sufficiently small (see, for example, [Reference FischerFis01, §7.2]). Since $\mu $ is minimal, $(s^{\pm \mu }, b_0(s))$ is a parametrization of ${\mathcal {E}}_x$ for some puncture x of a periodic curve with period dividing p.

We now show that the Puiseux series version of the main lemma (i.e., Lemma 5.2) is equivalent to the main lemma itself (i.e., Lemma 3.1). Given a puncture x with local parametrization $(s^{\pm \mu }, b_x(s))$ , any meromorphic function $w:{\mathcal {E}}_x \to \overline {\mathbb {C}}$ has a Laurent series expansion in the local parameter s. That is, $w$ is an element of $\mathbb {C}((s))$ . Under the morphism determined by $s \mapsto t^{1/\mu }$ between $\mathbb {C}((s))$ and $\mathbb {C}((t^{1/\mu }))$ , the order at $0$ is divided by $\mu $ . This simple observation leads to the equivalence between the two versions of the main lemma.

For simplicity, we use partial derivatives notation. In particular, the derivatives involved in (5.1) will be denoted by $\partial _\beta \unicode{x3c9} _p (\beta _0)$ and $\partial _z \varphi ^{\circ p-1}_{\beta _0} (\infty )$ .

5.2 A complete field

Recall that $\mathbb {C} \langle \langle t \rangle \rangle $ is endowed with the non-Archimedean absolute value

$$ \begin{align*} | z |_{\mathrm{o}} := \exp(-\mathrm{ord}_0\ z).\end{align*} $$

The metric induced by $| z |_{\mathrm {o}}$ in $\mathbb {C} \langle \langle t \rangle \rangle $ is particularly suited to understanding the orders of dynamically defined quantities around the punctures of ${\mathcal {S}}_p$ . However, endowed with this metric, $\mathbb {C} \langle \langle t \rangle \rangle $ is not complete. We let $\mathbb {L}$ be the field obtained as the completion of $\mathbb {C} \langle \langle t \rangle \rangle $ . It follows that $\mathbb {L}$ is also algebraically closed (see, for example, [Reference CasselsCas86]). Moreover, each element z of $\mathbb {L}$ can be represented by a series of the form

$$ \begin{align*}z = \sum_{n \geq 0} c_n t^{\unicode{x3bb}_n},\end{align*} $$

where $\unicode{x3bb} _n \in \mathbb {Q}$ and $\unicode{x3bb} _n \to +\infty $ as $n \to +\infty $ . Both $\mathrm {ord}_0\ z$ and $| z |_{\mathrm {o}}$ naturally extend to $\mathbb {L}$ . The value group $| \mathbb {L}^\times |_{\mathrm {o}}$ of $\mathbb {L}$ is $\exp (\mathbb {Q})$ .

We regard rational maps $\varphi \in \mathbb {L}(z)$ as dynamical systems acting on the projective line $\mathbb {P}^1_{\mathbb {L}}$ over $\mathbb {L}$ , and sometimes on the Berkovich projective line $\mathbb {P}^{1, {an}}$ over $\mathbb {L}$ . For our purpose here, passing from $\mathbb {C} \langle \langle t \rangle \rangle $ to $\mathbb {L}$ is not essential. However, non-Archimedean dynamics is better understood with the aid of Berkovich spaces over complete fields. For background about non-Archimedean dynamics see [Reference BenedettoBen19, Reference Baker and RumelyBR10]. In fact, we intensively employ the full description for quadratic rational dynamics over $\mathbb {L}$ given by the author in [Reference KiwiKiw14]. Although we summarize the main facts about quadratic dynamics in §5.3, for a detailed exposition we refer the reader to [Reference KiwiKiw14].

In §§5.2.1–5.2.3 we summarize basic facts about non-Archimedean analysis that will be used frequently in the rest of the paper.

5.2.1 Spherical metric

The projective line $\mathbb {P}^1_{\mathbb {L}}$ is identified with $\mathbb {L} \cup \{ \infty \}$ via ${[z:1]\mapsto z}$ and $[1:0] \mapsto \infty $ . In $\mathbb {P}^1_{\mathbb {L}}$ we work with the spherical metric which, for $z, w \in \mathbb {L}$ , is given by

$$ \begin{align*}\mathrm{dist}_s (z,w) := \begin{cases} | z-w |_{\mathrm{o}} & \text{ if } | z |_{\mathrm{o}},| w |_{\mathrm{o}} \le 1,\\ | 1/z - 1/w |_{\mathrm{o}} & \text{ if } | z |_{\mathrm{o}},| w |_{\mathrm{o}}> 1,\\ 1 & \text{ otherwise}. \end{cases} \end{align*} $$

It naturally extends to $z=\infty $ or $w=\infty $ .

In dynamical space, we will compute the size of derivatives in the spherical metric and denote them by $\| \cdot \|$ . For example, $\|\partial _z \varphi _\beta (\infty )\|=| \beta |_{\mathrm {o}}$ , for all $\beta \neq 0$ . In parameter space, we work in the standard coordinate of $\mathbb {L}$ . For example, $\| \partial _\beta \varphi _\beta (z) \|$ is $| \partial _\beta (1/\varphi _\beta (z)) |_{\mathrm {o}}$ if $| \varphi _\beta (z) |_{\mathrm {o}}>1$ . In this language, the main lemma states that for $n=p$ ,

(5.2) $$ \begin{align} \|\partial_\beta \unicode{x3c9}_n(\beta_0)\| = \|\partial_z \varphi_{\beta_0}^{\circ n-1}(\infty)\| | \tau |_{\mathrm{o}}^2. \end{align} $$

Our strategy will be to prove (5.2) for an increasing sequence of values of n, finishing with $n=p$ .

5.2.2 Reduction: from Puiseux series to $\mathbb {C}$

Often we will pass from dynamics over $\mathbb {L}$ to dynamics over $\mathbb {C}$ via reduction. Intuitively, reduction amounts to replacing t by $0$ . More precisely, the set of elements $z= \tilde {z}+o(1) \in \mathbb {L}$ is the ring of integers

$$ \begin{align*}\mathfrak{O} := \{ z \in \mathbb{L} : | z |_{\mathrm{o}}\le 1 \}.\end{align*} $$

The elements of its unique maximal ideal

$$ \begin{align*}\mathfrak{M}:= \{ z \in \mathbb{L} : | z |_{\mathrm{o}}<1 \}\end{align*} $$

are of the form $z=o(1)$ . The quotient $\mathfrak {O}/\mathfrak {M}$ is canonically identified with $\mathbb {C}$ via

$$ \begin{align*}\tilde{z} + \mathfrak{M} \mapsto \tilde{z},\end{align*} $$

for $\tilde {z} \in \mathbb {C} \subset \mathbb {L}$ . The quotient map naturally extends to

$$ \begin{align*}\mathrm{red} : \mathbb{P}^1_{\mathbb{L}} \to \mathbb{P}^1_{\mathbb{C}}\end{align*} $$

by declaring $\mathrm {red}(z) =\infty $ when $| z |_{\mathrm {o}}> 1$ . This map is called the reduction map. Note that each spherical open ball of radius $1$ maps onto a point under reduction. The obvious inclusion $\mathbb {P}^1_{\mathbb {C}} \hookrightarrow \mathbb {P}^1_{\mathbb {L}}$ is a right inverse of $\mathrm {red}$ .

Although we mostly work with non-homogeneous coordinates, to understand the effect of reduction on rational maps it is convenient to work in homogeneous coordinates. Any rational map $\varphi : \mathbb {P}^1_{\mathbb {L}} \to \mathbb {P}^1_{\mathbb {C}}$ can be written as a quotient of degree d homogeneous polynomials $P, Q$ with coefficients in $\mathfrak {O}$ and with at least one coefficient of absolute value $1$ . The polynomials $\tilde {P}, \tilde {Q}$ obtained after reducing the coefficients of P and Q might have a common factor, say H. Then there exist relatively prime polynomials $\widehat {P},\widehat {Q}$ with complex coefficients and a maximal degree common divisor H of $\tilde {P}, \tilde {Q}$ such that $\tilde {P} = H \cdot \widehat {P}$ and $\tilde {Q} = H \cdot \widehat {Q}$ . We say that

$$ \begin{align*} \begin{array}{rccl} \mathrm{red}\ \varphi:&{\mathbb{P}}^1_{\mathbb{C}}&\to&{\mathbb{P}}^1_{\mathbb{C}} \\ {}& [z:w]&\mapsto&[{\widehat{P}}(z,w),{\widehat{Q}}(z,w)] \end{array} \end{align*} $$

is the reduction of $\varphi $ . When convenient we denote $\mathrm {red}(z)$ by $\tilde {z}$ and $\mathrm {red}\ \varphi $ by $\tilde {\varphi }$ .

The reduction map semiconjugates $\varphi : \mathbb {P}^1_{\mathbb {L}} \to \mathbb {P}^1_{\mathbb {L}}$ with $\mathrm {red}\ \varphi : \mathbb {P}^1_{\mathbb {C}} \to \mathbb {P}^1_{\mathbb {C}}$ for all $z \in \mathbb {P}^1_{\mathbb {L}}$ outside a finite collection of disjoint open balls of spherical radius $1$ . In fact,

$$ \begin{align*}(\mathrm{red}\ \varphi)(\tilde{z}) = \mathrm{red}(\varphi(z))\end{align*} $$

for all $z \in \mathbb {P}^1_{\mathbb {L}}$ such that $H(\tilde {z}) \neq 0$ . Moreover, if $\deg \tilde {\varphi } \ge 1$ , then $\varphi $ maps the ball $\mathrm {red}^{-1} (\tilde {z})$ onto the ball $\mathrm {red}^{-1}(\tilde {\varphi }(\tilde {z}))$ with degree $\deg _{\tilde {z}} \tilde {\varphi }$ , provided $H(\tilde {z})\neq 0$ .

5.2.3 Action on balls and annuli

A ball B in $\mathbb {L}$ is a set of the form

$$ \begin{align*}\{ z \in \mathbb{L} : | z-z_0 |_{\mathrm{o}} < r \}\quad \text{or}\quad \{ z \in \mathbb{L} : | z-z_0 |_{\mathrm{o}} \le r \},\end{align*} $$

for some $z_0 \in \mathbb {L}$ and some $r>0$ . In the former case we say that B is an open ball, in the latter it is a closed ball. All balls are topologically open and closed. If $r \notin |\mathbb {L}^\times | = \exp (\mathbb {Q})$ , then the open ball and closed ball of radius r around $z_0$ coincide. If $r \in |\mathbb {L}^\times |$ , then we say that the corresponding balls are rational open or rational closed balls.

An analytic function $\varphi : \mathfrak {O} \to \mathfrak {O}$ is a power series $\sum c_i z^i$ with coefficients in $\mathfrak {O}$ such that $| c_i |_{\mathrm {o}} \to 0$ ; equivalently, the series converges for all $z \in \mathfrak {O}$ . The reduction ${\tilde {\varphi } = \sum \tilde {c}_i \tilde {z}^i \in \mathbb {C}[\tilde {z}]}$ is a polynomial. Moreover, $\deg \tilde {\varphi } \ge 1$ if and only if $\varphi $ is onto. In this case, $\deg \varphi = \deg \tilde {\varphi }$ is the number of preimages under $\varphi $ , counting multiplicities, of points in $\mathfrak {O}$ . Moreover, $\varphi $ maps the maximal open ball $\tilde {z} + \mathfrak {M}$ onto $\tilde {\varphi }(\tilde {z}) + \mathfrak {M}$ with degree $\deg _{\tilde {z}} \tilde {\varphi }$ . Note that $\deg \varphi $ is the number of critical points of $\varphi $ , counting multiplicities, plus $1$ . This is true since the residue field $\mathfrak {O}/\mathfrak {M}=\mathbb {C}$ has characteristic zero.

We identify $\mathbb {P}^1_{\mathbb {L}}$ with $\mathbb {L} \cup \{\infty \}$ . If $\varphi : \mathbb {P}^1_{\mathbb {L}} \to \mathbb {P}^1_{\mathbb {L}}$ has no poles in a ball B, say of radius r, then $\varphi (B)$ is a ball of the same type (rational, open, closed), say of radius $r'$ . Moreover, the Schwarz lemma holds; that is, for all $z \in B$ ,

$$ \begin{align*}\bigg| \dfrac{d\varphi}{dz}(z) \bigg|_{\mathrm{o}}\le \dfrac{r'}{r},\end{align*} $$

with equality if and only if $\varphi : B \to \varphi (B)$ is a bijection; equivalently,

$$ \begin{align*}| \varphi(z)-\varphi(z') |_{\mathrm{o}}= \dfrac{r'}{r} | z-z' |_{\mathrm{o}}\end{align*} $$

for all $z,z' \in B$ . Furthermore, since the residue field has characteristic zero, the mean value theorem also holds: if $\varphi :B \to \mathbb {L}$ and $| \varphi '(z) |_{\mathrm {o}} \le C$ for all $z \in B$ , then for all $z,z' \in ~B$ ,

$$ \begin{align*}| \varphi(z)-\varphi(z') |_{\mathrm{o}} \le C | z-z' |_{\mathrm{o}}.\end{align*} $$

In general, a (respectively, rational, open) closed ball B in $\mathbb {P}^1_{\mathbb {L}}$ containing $\infty $ is the complement of a (respectively, rational, closed) open ball in $\mathbb {L}$ . Given a rational map $\varphi : \mathbb {P}^1_{\mathbb {L}} \to \mathbb {P}^1_{\mathbb {L}}$ and a ball B, then $\varphi (B)$ is a ball of the same type or $\varphi (B)=\mathbb {P}^1_{\mathbb {L}}$ . If B is a ball and $\varphi (B)$ is a ball, then $\varphi : B \to \varphi (B)$ has a well-defined degree which is the number of critical points in B, counting multiplicities, plus $1$ . If, moreover, B is a closed rational ball, then each maximal open ball in B maps onto a maximal open ball in $\varphi (B)$ .

If $\mathbb {L} \supset B \supset B'$ are nested balls, then $A=B\setminus B'$ is an annulus. Its modulus is $\mathrm {mod}(A):=\log \mathrm {diam}(B) - \log \mathrm {diam} (B')$ . These definitions easily generalize to $\mathbb {P}^1_{\mathbb {L}}$ . If A is an annulus and $\varphi : \mathbb {P}^1_{\mathbb {L}} \to \mathbb {P}^1_{\mathbb {L}}$ is a rational map such that $\varphi (A)$ is again an annulus, then $\varphi : A \to \varphi (A)$ has a well-defined degree $d_A\ge 1$ and $d_A \cdot \mathrm {mod}(A) = \mathrm {mod}(\varphi (A))$ .

5.3 Quadratic rational dynamics over $\mathbb {L}$ : limbs

In our family $\varphi _\beta $ we restrict our attention to certain subsets of parameter space that we call limbs, by analogy with complex dynamics. More precisely, given $\theta \in \mathbb {Q}/\mathbb {Z}$ of order $q \ge 2$ , we define the non-Archimedean limb ${\mathcal {L}}_\theta \subset \mathbb {L}$ by:

$$ \begin{align*} {\mathcal{L}}_{1/2} &:= \{ \beta = \gamma t^{-1} : | \gamma |_{\mathrm{o}}= 1 \}, \\ {\mathcal{L}}_\theta&:= \{ \beta = c_\theta + \gamma t : | \gamma |_{\mathrm{o}}\le 1 \} \quad \text{if } q \ge 3, \end{align*} $$

where $c_\theta := -(2 \cos \pi \theta )^{-2}.$

By Lemmas 4.5 and 5.1, limbs contain the parameters of interest to us. Indeed, any $\beta _0$ for which $\unicode{x3c9} $ is periodic of period $p \ge 3$ is contained in some limb. In this section, we introduce basic results about the dynamics of $\varphi _\beta $ with a focus on parameters that lie in a limb.

Below, we show that all maps $\varphi _\beta $ in a limb have an invariant Rivera domain $U_0$ . Although Rivera domains are easier to understand in Berkovich spaces, we will keep our discussion in the classical line $\mathbb {P}^1_{\mathbb {L}}$ . In this context, an invariant Rivera domain $U_0$ is the complement of finitely many pairwise disjoint rational closed balls such that $\varphi : U_0 \to U_0$ is a bijection and $U_0$ is maximal with respect to these properties.

To describe the dynamics and the invariant domain $U_0$ of $\varphi _\beta $ when $\beta $ lies in an order q limb ${\mathcal {L}}_\theta $ , we define q closed balls $B_0, \ldots , B_{q-1}$ , subscripts mod q, and a maximal open ball $D_0$ contained in $B_0$ . We do not make the dependence of $B_j$ on $\theta $ explicit. The open ball $D_0$ is independent of $\beta $ , except in the case $q=2$ .

Given $\beta =\gamma t^{-1} \in {\mathcal {L}}_{1/2}$ , let

$$ \begin{align*} B_0 &:= \{ z:| z |_{\mathrm{o}} \le 1 \}, \\ B_1 &:= \{ z:| z |_{\mathrm{o}} \ge | t |_{\mathrm{o}}^{-1}\}, \\ D_0 &:= \{ z:| z-\gamma^{-1} |_{\mathrm{o}}<1 \}. \end{align*} $$

Given $\theta \in \mathbb {Q} / \mathbb {Z}$ of order $q \ge 3$ , consider $\beta = c_\theta + o(1) \in {\mathcal {L}}_\theta $ . Recall that $\unicode{x3c9} _j(c_\theta )$ denotes the jth iterate of $\unicode{x3c9} =0$ under $\varphi _{c_\theta }$ . Let

$$ \begin{align*} B_j &:= \{ z: | z - \unicode{x3c9}_j(c_\theta) |_{\mathrm{o}} \le | t |_{\mathrm{o}} \} \quad \text{if } j=0 \text{ or } 2 \le j < q, \\ B_1 &:= \{ z: | z |_{\mathrm{o}} \ge | t |_{\mathrm{o}}^{-1}\}, \\ D_0 &:= \{ z: | z+c_{\theta}^{-1}t |_{\mathrm{o}} < | t |_{\mathrm{o}} \}. \end{align*} $$

Note that the critical points of $\varphi _\beta $ lie in $B_0$ and the critical values in $B_1$ .

Proposition 5.4. Consider a parameter $\beta $ in a limb ${\mathcal {L}}_\theta $ of order q. Let

$$ \begin{align*}U_0 := \mathbb{P}^1_{\mathbb{L}} \setminus \bigcup_{j=0}^{q-1} B_j.\end{align*} $$

Then the following assertions hold:

1. (1) $\varphi _\beta : U_0 \to U_0$ is a bijection;

2. (2) if $j\neq 0$ , then $\varphi _\beta : B_j \to B_{j+1}$ is a bijection;

3. (3) $\varphi _\beta : B_0 \setminus D_0 \to B_1$ has degree $2$ ;

4. (4) $\varphi _\beta : D_0 \to \mathbb {P}^1_{\mathbb {L}} \setminus B_1$ is a bijection.

Proof. We assume that $q \ge 3$ . The case $q=2$ follows from a similar argument.

Consider a parameter $\beta \in {\mathcal {L}}_\theta $ . To simplify notation write $c:=c_\theta $ . Hence, $\varphi _c$ is the map corresponding to $\beta =c$ . Also,

$$ \begin{align*}M(\tilde{z}):=1 + \dfrac{c}{\tilde{z}} =\mathrm{red}\ \varphi_\beta (\tilde{z}) \end{align*} $$

is an order q Möbius transformation.

The semiconjugacy between $\varphi _\beta $ and M, furnished by reduction, holds for all $z \notin \mathfrak {M}$ . Hence, the open ball $\tilde {z} + \mathfrak {M}$ of radius $1$ is mapped isometrically (in the spherical metric) onto $M(\tilde {z}) + \mathfrak {M}$ , for all $\tilde {z} \in \overline {\mathbb {C}} \setminus \{0\}$ . In particular, $\unicode{x3c9} _j(\beta ) \notin \mathfrak {M}$ , for $1\le j< q$ . Furthermore, a ball of spherical radius $| t |_{\mathrm {o}}$ around $\unicode{x3c9} _j (\beta )$ maps onto a ball with the same spherical radius around $\unicode{x3c9} _{j+1} (\beta )$ , for all $1 \le j < q$ .

We postpone the proof of (1). To prove (2), let $\gamma \in \mathfrak {O}$ be such that for $\beta = c + \gamma t \in {\mathcal {L}}_\theta $ . We claim that, for all $2\le j \le q$ ,

(5.3) $$ \begin{align} \unicode{x3c9}_j (\beta) &= \unicode{x3c9}_j(c) + O(t). \end{align} $$

Indeed, this is trivially true for $j=2$ and we proceed by induction. Suppose it is true for some j such that $2 \le j< q$ . Then

$$ \begin{align*}| \varphi_\beta(\unicode{x3c9}_j(\beta))-\varphi_\beta(\unicode{x3c9}_j(c)) |_{\mathrm{o}} = | \unicode{x3c9}_j(\beta) - \unicode{x3c9}_j(c) |_{\mathrm{o}} \le | t |_{\mathrm{o}},\end{align*} $$

since $\varphi _\beta $ is an isometry on open balls of radius $1$ disjoint from $\mathfrak {M}$ . Moreover,

$$ \begin{align*}| \varphi_\beta(\unicode{x3c9}_j(c))-\varphi_c(\unicode{x3c9}_j(c)) |_{\mathrm{o}}= \bigg| \dfrac{t \gamma}{\unicode{x3c9}_j(c)} \bigg|_{\mathrm{o}} \le | t |_{\mathrm{o}}.\end{align*} $$

From the strong triangle inequality, (5.3) holds for $j+1$ . Therefore, (2) holds for ${1 \le j \le q-2}$ . To prove (2) for $j=q-1$ , we must show that $\unicode{x3c9} _q(c)=\unicode{x3c9} +O(t)$ . In fact, a straightforward induction shows that for $2 \le j \le q$ ,

$$ \begin{align*} \unicode{x3c9}_j (c) &= M^{\circ j-2} (1) + O(t). \end{align*} $$

Since $ M^{\circ q-2} (1) =0=\unicode{x3c9} $ , assertion (2) holds.

For (3) and (4), note that

$$ \begin{align*}h(\tilde{z}):= \mathrm{red} ( {t} {\varphi_\beta (zt)} ) = \dfrac{c\tilde{z}+1}{\tilde{z}^2}.\end{align*} $$

It follows that $D_0 = \{z: z= -c^{-1} t + o(t)\}$ maps bijectively onto $\mathfrak {M}$ by $t \varphi _\beta $ , which proves (4). For all $\tilde {z} \neq -c^{-1}$ , the open ball of the form $\tilde {z} t + o(t)$ is mapped by $\varphi _\beta $ onto the open ball of spherical radius $| t |_{\mathrm {o}}$ centered at $h(\tilde {z})/t$ with degree $\deg _{\tilde {z}} h$ , which yields (3).

To finish, we deduce (1). First observe that every element of $\mathbb {P}^1_{\mathbb {L}} \setminus U_0$ has two preimages, counting multiplicities, in $\mathbb {P}^1_{\mathbb {L}} \setminus U_0$ . Thus, $\varphi _\beta (U_0) \subset U_0$ . By (4), a point $z \in U_0$ has exactly one preimage in $D_0$ . By (2) and (3), no preimage of z is contained in ${B_1, \ldots , B_{q-1}, B_0\setminus D_0}$ . Hence, z has exactly one preimage in $U_0$ . That is, $\varphi _\beta (U_0) = U_0$ and $\varphi _\beta $ is injective in  $U_0$ .

We gather together basic consequences about the action of $\varphi _\beta $ on balls and annuli.

We omit the proof of the corollary which is an application of §5.2.3 and Proposition 5.4. To prove (4) it is not difficult to check that $U_0$ is maximal with respect to the property of being a complement of finitely many balls such that $\varphi _\beta :U_0\to U_0$ is bijective.

5.4 Parabolic rescaling

The balls $B_0, \ldots , B_{q-1}$ correspond to a ‘parabolic rescaling limit’. That is, according to [Reference KiwiKiw14, Theorems 1 and 2], after a change of coordinates mapping $B_0$ onto the ring of integers $\mathfrak {O}$ , the map $\varphi ^{\circ q}_\beta $ reduces to a complex quadratic rational map with a parabolic fixed point. We will need to be precise about the dependence of the complex parabolic map on the parameter $\beta $ . Although one could implement a non-Archimedean version of the computations leading to Proposition 4.4, for simplicity, we prefer to employ this complex dynamics result.

To write down the dependence on $\beta $ , we introduce parametrizations for the limbs and convenient coordinate changes. Given $\theta \in \mathbb {Q}/\mathbb {Z}$ of order $q \ge 3$ , we let

$$ \begin{align*} \unicode{x3bb}_\theta(\gamma) & := c_\theta + \gamma t \quad\text{for } |\gamma|\le 1, \\ L_\theta (z) &:= t^{-1} c_\theta z. \end{align*} $$

Also,

$$ \begin{align*} \unicode{x3bb}_{1/2}(\gamma) & := \dfrac{\gamma}{t} \quad \text {for } |\gamma|=1, \\ L_{1/2} (z)&:= \gamma z. \end{align*} $$

We abuse notation and consider the dependence of $L_{1/2}$ on $\gamma $ implicitly. Observe that $\unicode{x3bb} _\theta $ is the natural parametrization of the $\theta $ -limb ${\mathcal {L}}_\theta $ .

Consider

$$ \begin{align*} \begin{array}{cccc} \pi_0:&B_0& \to& \mathbb{P}^1_{\mathbb{C}}\\ & z & \to & \mathrm{red}(L_\theta(z)). \end{array} \end{align*} $$

Then, given $\beta \in {\mathcal {L}}_\theta $ , the map $\varphi _\beta ^{\circ q} : B_0 \setminus D_0 \to B_0$ is semiconjugate, via $\pi _0$ , to a complex quadratic rational map R with a parabolic fixed point [Reference KiwiKiw14, Theorems 1,2]. Moreover, the parabolic fixed point of R is at $\pi _0(U_0)=\infty $ with preimage $\pi _0(D_0)=-1$ and R has a critical point in $\pi _0 (\unicode{x3c9} ) =0$ . Hence, R is an element $R_v$ of the parabolic family (4.1) and

$$ \begin{align*}\pi_0 \circ \varphi_\beta^{\circ q} (z) = R_v \circ \pi_0(z),\end{align*} $$

for $z \in B_0 \setminus D_0$ . Equivalently,

$$ \begin{align*}R_v = \mathrm{red}(L_\theta \circ \varphi^{\circ q}_{\beta} \circ L^{-1}_\theta ),\end{align*} $$

where

$$ \begin{align*}v := \mathrm{red} (L_\theta (\unicode{x3c9}_q(\beta))).\end{align*} $$

The following lemma says that the assignment $\beta \mapsto v$ induces an isomorphism of complex affine lines.

Lemma 5.6. Given $\theta \in \mathbb {Q}/\mathbb {Z}$ of order $q \ge 3$ (respectively, $q=2$ ), there exists a non-constant function $v_\theta (\tilde {\gamma }) = A \tilde {\gamma } +B \in \mathbb {C}[\tilde {\gamma }]$ (respectively, $v_\theta (\tilde {\gamma })={\tilde {\gamma }}$ ) such that, for all $\gamma \in \mathfrak {O}$ (respectively, $\gamma \in \mathfrak {O} \setminus ~\mathfrak {M}$ ),

$$ \begin{align*}\mathrm{red}(L_\theta \circ \varphi^{\circ q}_{\unicode{x3bb}_\theta (\gamma)} \circ L^{-1}_\theta ({z})) = R_{v(\tilde{\gamma})}(\tilde{z}) \end{align*} $$

for all $z \in B_0 \setminus D_0$ or equivalently for all $\tilde {z} \in \mathbb {C}\setminus \{-1\}$ .

Proof. When $\theta =1/2$ , a straightforward direct computation proves the assertion.

If $\theta $ has order $q \ge 3$ , then the function

$$ \begin{align*}\unicode{x3c9}_q : {\mathcal{L}}_\theta \to B_0\end{align*} $$

is well defined. Therefore,

$$ \begin{align*}L_\theta (\unicode{x3c9}_q(\unicode{x3bb}_\theta (\gamma))) \in \mathfrak{O},\end{align*} $$

and

$$ \begin{align*}w(\gamma):=\mathrm{red}\ L_\theta (\unicode{x3c9}_q(\unicode{x3bb}_\theta (\gamma)))\end{align*} $$

only depends on ${\tilde {\gamma }}$ . That is, $w(\gamma ) = v_\theta ({\tilde {\gamma }})$ for some $v_\theta ({\tilde {\gamma }}) \in \mathbb {C}[{\tilde {\gamma }}]$ . We must show that ${\tilde {\gamma }} \to v_\theta ({\tilde {\gamma }})$ is an affine isomorphism.

According to Proposition 4.4, there exists a non-constant complex affine map ${{\tilde {\gamma }} \mapsto A{\tilde {\gamma }} + B}$ such that if $\unicode{x3bb} _\theta (\gamma )= c_\theta + {\tilde {\gamma }} t$ , then

$$ \begin{align*}A{\tilde{\gamma}} + B = \lim_{t\to 0} t^{-1} c_\theta \omega_q(c_\theta+{\tilde{\gamma}} t). \end{align*} $$

Since reduction coincides with the limit as $t \to 0$ ,

$$ \begin{align*} A{\tilde{\gamma}} + B = \lim_{t\to 0} t^{-1} c_\theta \omega_q(c_\theta+{\tilde{\gamma}} t)=\mathrm{red} (L_\theta (\unicode{x3c9}_q(c_\theta + {\tilde{\gamma}} t)) = v_\theta ({\tilde{\gamma}}). \\[-42pt] \end{align*} $$

For a parameter $\beta $ in an order q limb, we say that $\unicode{x3c9} $ has a non-central return if ${\unicode{x3c9} _{q\ell } (\beta ) \in D_0}$ for some $\ell \ge 1$ . In this case, we say that $q\ell _0$ is the first non-central return time if $\ell _0$ is minimal such that $\unicode{x3c9} _{q\ell _0} (\beta ) \in D_0$ .

A parameter without non-central returns is called a satellite parameter.

5.5 Preliminary bounds

The rest of the paper is devoted to the proof of the main Lemma 5.2. Here we list some basic preliminary bounds. Specifically, for future reference, we record the size of $\partial _z \varphi _\beta (z)$ and $\partial _\beta \varphi _\beta (z)$ . Then, in §5.6, our work towards proving (5.2) for $n=p$ will start (and finish) at $n=p$ , for satellite parameters, and at $n=q \ell _0+1$ , for parameters with non-central returns.

The critical points of $\varphi _\beta $ are $\unicode{x3c9} =0$ and $\unicode{x3c9} ' = -2t/\beta $ (respectively, $-2t^{-1}/\beta $ ) if ${| \beta |_{\mathrm {o}}\le 1}$ (respectively, if ${| \beta |_{\mathrm {o}}> 1}$ ). Assuming that $\beta $ lies in some limb ${\mathcal {L}}_\theta $ , denote by $D^{\prime }_0$ the maximal open ball contained in $B_0$ such that $\unicode{x3c9} ' \in D_0'$ . Let

$$ \begin{align*}B^{\prime}_0:=B_0 \setminus D_0'\end{align*} $$

and

$$ \begin{align*}B_{1/2} := C_1 \cup \cdots \cup C_{q-1},\end{align*} $$

where $C_j$ for $1\le j < q$ are the disks from Corollary 5.5 (2). Note that $B_{1/2}$ is the union of $q-1$ balls that depend on $\beta $ , but we omit this dependence in the notation. If $\beta _0$ is such that $\unicode{x3c9} $ is periodic, then $\unicode{x3c9} _n(\beta _0) \in B_{1/2}$ whenever $\unicode{x3c9} _n(\beta _0) \in D_0$ , for otherwise $\unicode{x3c9} _{n+1}(\beta _0) \in U_0$ .

Lemma 5.9. Let ${\mathcal {L}}_\theta $ be a limb of order $q \ge 2$ . Assume $\beta \in {\mathcal {L}}_\theta $ .

If $q=2$ , then

(5.4) $$ \begin{align} \|\partial_z \varphi_\beta(z)\| & = \begin{cases} | z |_{\mathrm{o}}\cdot | t |_{\mathrm{o}} & \text{ if } z \in B^{\prime}_0,\\ | t |_{\mathrm{o}}^{-1} & \text{ if } z \in B_1 \cup B_{1/2}; \end{cases} \end{align} $$

(5.5) $$ \begin{align} \|\partial_\beta \varphi_\beta (z)\|&= \begin{cases} | z |_{\mathrm{o}}^3 \cdot | t |_{\mathrm{o}}^2 & \text{ if } z \in B_0' ,\\ 1 & \text{ if } z \in B_{1/2}, \\ 1/| z |_{\mathrm{o}} & \text{ if } z \in B_1. \end{cases}\ \ \qquad \end{align} $$

If $q \ge 3$ , then

(5.6) $$ \begin{align} \|\partial_z \varphi_\beta(z)\| & = \begin{cases} | z |_{\mathrm{o}}\cdot | t |_{\mathrm{o}}^{-1} & \text{ if } z \in B^{\prime}_0 , \\ | t |_{\mathrm{o}}^{-2} & \text{ if } z \in B_{1/2}, \\ 1 & \text{ if } z \in B_1 \cup \cdots \cup B_{q-1}; \end{cases} \end{align} $$

(5.7) $$ \begin{align} \|\partial_\beta \varphi_\beta (z)\|&= \begin{cases} | z |_{\mathrm{o}}^3 \cdot | t |_{\mathrm{o}}^{-2} & \text{ if } z \in B_0' , \\ | t |_{\mathrm{o}}^{-1} & \text{ if } z \in B_{1/2}, \\ 1/| z |_{\mathrm{o}} & \text{ if } z \in B_1, \\ 1 & \text{ if } z \in B_2 \cup \cdots \cup B_{q-1}. \end{cases} \end{align} $$

Denote by $\mathrm {diam} (B)$ the spherical diameter of a ball contained in $\mathfrak {O}$ or in $\mathbb {P}^1_{\mathbb {L}} \setminus \mathfrak {O}$ . Observe that if $z \in B_j$ for some $j\neq 0$ , then

$$ \begin{align*}\| \partial_z \varphi_\beta (z) \| = \dfrac{\mathrm{diam} (B_{j+1})}{\mathrm{diam} (B_j)},\end{align*} $$

subscripts mod q. If $z \in B_0'$ and $| z |_{\mathrm {o}} = \mathrm {diam} (B_0)$ , then

$$ \begin{align*}\| \partial_z \varphi_\beta (z) \| = \dfrac{\mathrm{diam} (B_{1})}{\mathrm{diam} (B_0)}.\end{align*} $$

This can be checked directly as an application of Lemma 5.9 or from the Schwarz lemma.

The semiconjugacy $\mathrm {red}(L_\theta (z))$ furnished by Lemma 5.6 maps the open ball $D_0'$ onto a critical point $\omega '$ which lies in the basin of the multiple fixed point, under the corresponding parabolic map. Hence, the orbit of $\unicode{x3c9} $ is disjoint from $D_0'$ and every point in $D_0'$ has an infinite orbit.

5.6 Satellite and first non-central return

We now prove the initial cases of (5.2) according to whether a first non-central return exists or not.

Proof. If $\beta _0$ is a satellite parameter, it will be convenient to let $\ell _0=p/q$ . Otherwise, the first non-central return is $q\ell _0$ as is in the statement of the lemma.

We consider first the case in which $\beta _0$ lies in the $1/2$ -limb. In this limb, for $| \gamma |_{\mathrm {o}}=1$ , let $\unicode{x3bb} (\gamma ):= \gamma /t$ . Then $\unicode{x3bb} (\gamma _0) = \beta _0$ for $\gamma _0 := t \beta _0$ . In view of Lemma 5.6, for all $z \in B_0 \setminus D_0$ ,

$$ \begin{align*}\mathrm{red} (\gamma \varphi^{\circ 2}_{\unicode{x3bb}(\gamma)} (z/\gamma)) = R_{\tilde{\gamma}}(\tilde{z}).\end{align*} $$

Hence,

$$ \begin{align*} \mathrm{red}(\unicode{x3c9}_{2 \ell_0}(\beta_0) + \gamma_0 \partial_\beta \unicode{x3c9}_{2 \ell_0}(\beta_0) t^{-1}) &= \mathrm{red}\bigg(\dfrac{d}{d\gamma}(\gamma \unicode{x3c9}_{2\ell_0}(\unicode{x3bb}(\gamma)))|_{\gamma=\gamma_0} \bigg) \\ & = \dfrac{d}{d {\tilde{\gamma}}} \mathrm{red} (\gamma \unicode{x3c9}_{2 \ell_0}(\unicode{x3bb}(\gamma))|_{{\tilde{\gamma}}={\tilde{\gamma}}_0} \\ &= \dfrac{d}{d{\tilde{\gamma}}} R^{\circ \ell_0}_{{\tilde{\gamma}}}(0)|_{{\tilde{\gamma}}={\tilde{\gamma}}_0}\\ &= C \neq 0. \end{align*} $$

The first line is an immediate consequence of the product and chain rules, taking into account that $\beta =\gamma /t$ . Reduction and derivatives commute in $B_0 \setminus D_0$ , which yields the second line. Reduction and iteration also commute in $B_0 \setminus D_0$ , thus the third line follows. Finally, note that $R^{\circ \ell _0}_{{\tilde {\gamma }}_0} (0) =0$ or $-1$ , so the last line is a consequence of Lemma 4.6.

If $2 \ell _0 =p$ , then $\unicode{x3c9} _{2 \ell _0} (\beta _0) =0$ . We conclude that

$$ \begin{align*}\| \partial_{\beta} \unicode{x3c9}_{2 \ell_0} (\beta_0) \| = | \partial_{\beta} \unicode{x3c9}_{2 \ell_0} (\beta_0) |_{\mathrm{o}} = | t |_{\mathrm{o}}.\end{align*} $$

If $2 \ell _0 <p$ , then

$$ \begin{align*} \unicode{x3c9}_{2 \ell_0} (\beta_0) &=1/{\tilde{\gamma}}_0 + o(1),\\ \partial_\beta \varphi_\beta (\unicode{x3c9}_{2 \ell_0} (\beta_0)) &= {\tilde{\gamma}}_0 + o(1),\\ \partial_z \varphi_{\beta_0}(\unicode{x3c9}_{2 \ell_0} (\beta_0)) & = -\beta_0 \cdot {\tilde{\gamma}}_0^2 - 2 {\tilde{\gamma}}_0^3/t + o(t^{-1})\\ & = {\tilde{\gamma}}_0^3/t + o(t^{-1}). \end{align*} $$

Therefore,

$$ \begin{align*} \partial_\beta \unicode{x3c9}_{2 \ell_0 + 1} (\beta_0) &= \partial_\beta \varphi_\beta (\unicode{x3c9}_{2 \ell_0}(\beta_0)) + \partial_z \varphi_\beta(\unicode{x3c9}_{2 \ell_0}) \partial_\beta \unicode{x3c9}_{2 \ell_0} ({\beta}_0)\\ &= {{\tilde{\gamma}}_0} + \dfrac{{\tilde{\gamma}}_0^3}{t} \cdot t \dfrac{C-1/{\tilde{\gamma}}_0}{{\tilde{\gamma}}_0} +o(1)\\ &={\tilde{\gamma}}_0^2 C +o(1). \end{align*} $$

Hence,

$$ \begin{align*}\|\partial_\beta \unicode{x3c9}_{2 \ell_0 + 1} (\beta_0)\| = | \partial_\beta \unicode{x3c9}_{2 \ell_0 + 1} (\beta_0) |_{\mathrm{o}}=1,\end{align*} $$

since $\unicode{x3c9} _{2 \ell _0 + 1} (\beta _0) \in B_0$ .

By (5.4),

$$ \begin{align*}\| \partial_z \varphi^{2 \ell_0-1}_{\beta_0} (\infty) \| = | t |_{\mathrm{o}}^{-1},\end{align*} $$

and the lemma follows for $q=2$ .

Now consider the case in which $q \ge 3$ . For $\gamma \in \mathfrak {O}$ , let $\unicode{x3bb} (\gamma ) := c_\theta + \unicode{x3bb} t$ be the standard parametrization of ${\mathcal {L}}_\theta $ . Then $\unicode{x3bb} (\gamma _0) = \beta _0$ for $\gamma _0 := (\beta _0-c_\theta )/t$ . Recall that for all $z \in B_0\setminus D_0 $ ,

$$ \begin{align*}\mathrm{red} (t^{-1} c_\theta \varphi^{\circ q}_{\unicode{x3bb}(\gamma)} (z/t^{-1} c_\theta)) = R_{v({\tilde{\gamma}})}(\tilde{z})\end{align*} $$

where, $v({\tilde {\gamma }})$ is a $\mathbb {C}$ -affine automorphism (Lemma 5.6). As above, the chain rule and the commutative properties of derivatives, reduction, and iteration yield

$$ \begin{align*} \mathrm{red}(c_\theta\partial_\beta \unicode{x3c9}_{q \ell_0}(\beta_0)) &= \mathrm{red}\bigg(\dfrac{d}{d\gamma}( t^{-1}c_\theta\unicode{x3c9}_{q\ell_0}(\beta(\gamma)))|_{\gamma=\gamma_0} \bigg) \\ & = \dfrac{d}{d {\tilde{\gamma}}} \mathrm{red} (t^{-1} c_\theta \unicode{x3c9}_{q \ell_0}(\beta(\gamma))|_{{\tilde{\gamma}}={\tilde{\gamma}}_0}\\ &= \dfrac{d}{d{\tilde{\gamma}}} R^{\circ \ell_0}_{v({\tilde{\gamma}})}(0)|_{{\tilde{\gamma}}={\tilde{\gamma}}_0}\\ &\neq 0. \end{align*} $$

If $q \ell _0 =p$ , then $\unicode{x3c9} _{q \ell _0} (\beta _0) =0$ . We conclude that

$$ \begin{align*}\| \partial_\beta \unicode{x3c9}_{q \ell_0} (\beta_0) \|=| \partial_\beta \unicode{x3c9}_{q \ell_0} (\beta_0) |_{\mathrm{o}} = 1.\end{align*} $$

If $q \ell _0 <p$ , then

$$ \begin{align*} \unicode{x3c9}_{q \ell_0} (\beta_0) &=t/c_\theta + o(t),\\ \partial_\beta \varphi_\beta (\unicode{x3c9}_{q \ell_0} (\beta_0)) &= c_\theta t^{-1} + o(t^{-1}),\\ \partial_z \varphi_{\beta_0}(\unicode{x3c9}_{q \ell_0} (\beta_0)) & = \beta_0 c^2_\theta t^{-2} - 2t c_\theta^3 t^{-3} + o(t^{-2})\\ & = c_\theta^3 t^{-2} +o(t^{-2}). \end{align*} $$

Therefore,

$$ \begin{align*} | \partial_\beta \unicode{x3c9}_{q \ell_0 + 1} (\beta_0) |_{\mathrm{o}} &= | \partial_\beta \varphi_\beta (\unicode{x3c9}_{q \ell_0}(\beta_0)) + \partial_z \varphi_{\beta_0}(\unicode{x3c9}_{q \ell_0}(\beta_0)) \partial_\beta \unicode{x3c9}_{q \ell_0} ({\beta}_0) |_{\mathrm{o}}\\ &=| \partial_z \varphi_{\beta_0}(\unicode{x3c9}_{q \ell_0}(\beta_0)) |_{\mathrm{o}}. \end{align*} $$

Moreover, $\unicode{x3c9} _{q\ell _0} (\beta _0) ,\unicode{x3c9} _{q\ell _0 + 1} (\beta _0) \notin B_1$ , so

$$ \begin{align*}\| \partial_\beta \unicode{x3c9}_{q \ell_0 + 1} (\beta_0) \|=\| \partial_z \varphi_{\beta_0}(\unicode{x3c9}_{q \ell_0}(\beta_0)) \|.\end{align*} $$

Furthermore,

$$ \begin{align*}\| \partial_z \varphi_{\beta_0}^{q \ell_0-1} (\infty) \| = 1,\end{align*} $$

by (5.6). For $q \ge 3$ , the lemma now follows from the chain rule.

5.7 Geometry of polynomial rescaling limits

Given $\beta $ in an order q limb ${\mathcal {L}}_\theta $ , the filled Julia set of $\varphi _\beta $ is

$$ \begin{align*}{\mathcal{K}}(\varphi_\beta) := \{ z \in \mathbb{P}^1_{\mathbb{L}} : \varphi^{\circ n}_\beta (z) \notin U_0 \text{ for all } n \ge 0 \}.\end{align*} $$

We are interested in the geometry of ${\mathcal {K}}(\varphi _\beta )$ around $\unicode{x3c9} $ for parameters with non-central returns.

Clearly, $X_0 \subset {\mathcal {K}}(\varphi _{\beta _0})$ and $q'$ divides p.

The size of the balls in the orbit of $X_0$ is controlled by the following result.

Lemma 5.13. Let $\beta $ be a parameter in an order q limb. Suppose that $X_0$ is a periodic rational closed ball of period $q'$ containing $\unicode{x3c9} $ and $q\ell _0$ is the first non-central return time of $\unicode{x3c9} $ . Given $0 \le j <q'$ , consider k such that

$$ \begin{align*}X_j := \varphi^{\circ j}_{\beta_0} (X_0) \subset B_k\end{align*} $$

and let

$$ \begin{align*}L_j := \log \dfrac{\mathrm{diam} (B_k)}{\mathrm{diam} (X_j)}.\end{align*} $$

Then

$$ \begin{align*}L_1 = \cdots =L_{q\ell_0}> L_j,\end{align*} $$

for all $j=q \ell _0 +1, \ldots , q'$ .

Given a parameter $\beta $ with a periodic ball $X_0$ as above, we will freely use the notation for $X_j$ and $L_j$ introduced in the lemma, subscripts mod $q'$ .

Proof. Given $X_j$ , let k be such that $X_j \subset B_k$ and

$$ \begin{align*}\Gamma_j := \{ B \subset \mathbb{P}^1_{\mathbb{L}} : B \text{ is a closed ball and } X_j \subset B \subsetneq B_{k} \}.\end{align*} $$

We use reverse interval notation in $\Gamma _j$ , which is totally ordered by inclusion. That is, $[A,B] \subset \Gamma _j$ consists of all the balls C such that $A \supset C \supset B$ . Moduli of annuli give a natural parametrization of $\Gamma _j$ by a subinterval of $\mathbb {R}$ (corresponding to the hyperbolic distance in $\mathbb {P}^{1, {an}} \setminus \mathbb {P}^1_{\mathbb {L}}$ ):

$$ \begin{align*} \begin{array}{ccc} \Gamma_j & \to & ]{}{-}\!\log \mathrm{diam} (B_{k}),\ {-}\!\log \mathrm{diam} (X_j)]{}\\ B & \mapsto & {-}\!\log \mathrm{diam}(B). \end{array} \end{align*} $$

We simply say that the length of $[A,B]$ is the modulus of $A \setminus B$ , so the parametrization preserves length.

We omit the subscript $\beta _0$ and simply write $\varphi $ . We will analyze the dynamics induced by $\varphi $ on $\cup \Gamma _j$ . Its action in $\Gamma _j$ will be also denoted by $\varphi $ . Recall that $\varphi $ maps $D_0 \setminus \cup C_j$ bijectively onto $U_0$ . Denote by D the open ball around $\unicode{x3c9} $ of diameter $\mathrm {diam} (B_0)$ .

If $X_j \subset D_0$ , then $X_j \subset C_k$ for some k and the portion $]B_0,C_k[$ of $\Gamma _j$ maps outside of $\cup \Gamma _j$ ; that is, $\varphi (B) \notin \cup \Gamma _j$ , for all $B\ {\in}\ ]B_0,C_k[$ . Moreover, $[C_k,X_j]$ maps bijectively onto $\Gamma _{j+1}\ {=}\ ]B_{k+1},X_{j+1}]$ . Furthermore, $\varphi : [C_k,X_j] \to \Gamma _{j+1}$ is length preserving since $C_k$ maps bijectively onto $B_k$ .

If $X_j \subset B_k$ for some $k \neq 0$ , then $\varphi : \Gamma _{j} \to \Gamma _{j+1}\ {=}\ ]B_{k+1},X_j]$ is a length-preserving bijection, since $\varphi $ maps $B_k$ onto $B_{k+1}$ bijectively.

The interval $]B_0,X_0]$ consists of balls contained in D and $\varphi : A \setminus B \to \varphi (A)\setminus \varphi (B)$ is a degree $2$ map if $]A,B] \subset\ ]B_0,X_0]$ . Therefore, $\varphi :\Gamma _0 \to \Gamma _1$ is a bijection that multiplies lengths by a factor of $2$ .

If $X_j \hspace{-0.5pt}\subset\hspace{-0.5pt} D$ for some $j\neq 0$ , then $\Gamma _j \hspace{-0.5pt}\cap\hspace{-0.5pt} \Gamma _0 \,{=}\,]B_0, S_j]$ for some $S_j$ . Hence, ${\varphi : \Gamma _j \to \Gamma _{j+1}}$ is a bijection that multiplies the length of subintervals of $]B_0,S_j]$ by a factor $2$ and preserves lengths of the subintervals of $[S_j,X_j]$ .

Note that $\varphi (\Gamma _j) \supset \Gamma _{j+1}$ for all j and equality holds only if $X_j$ is not contained in $D_0$ .

We are now ready to compare lengths of $\Gamma _j$ with $\Gamma _1$ . We claim that, for all $j \ge 1$ , there exists an injective piecewise continuous and length-preserving map $\psi : \Gamma _j \to \Gamma _1$ . Indeed, given $B \in \Gamma _j$ , let $n=n(B) \ge 0$ be minimal so that there exists $B' \in \Gamma _1$ with the property that $\varphi ^{\circ n}(B') =B$ . Such a number $n(B)$ always exists. In fact, B has a unique preimage C in $\Gamma _{j-1}$ . If $C \in \Gamma _0$ , then $B\in \Gamma _1$ and $n(B)=0$ . Otherwise, C maps injectively onto B. Taking preimages recursively, we obtain $n(B)$ and $B' \in \Gamma _1$ . Let $\psi : \Gamma _j \to \Gamma _1$ be the map that sends B to $B'$ . Note that $n(B) < j$ . Moreover, $n(B)$ is non-decreasing. Given $n \ge 0$ , let

$$ \begin{align*}I_n := \{ B \in \Gamma_j : n = n(B) \}.\end{align*} $$

If $B'=\psi (B)$ for some $B \in I_n$ , then $B', \varphi (B'), \ldots , \varphi ^{n-1}(B') \notin \Gamma _0$ , for otherwise $n(B) < n$ . Hence, $\psi : I_n \to \Gamma _1$ preserves length. Moreover, $L_1 \le L_j$ , since from the definition of $n(B)$ we have that $\psi (I_n) \cap \psi (I_m) = \emptyset $ if $n \neq m$ .

If $n(B)$ is constant, say n, in $\Gamma _j$ , then $\varphi ^{\circ n}: \Gamma _1 \to \Gamma _j$ is a bijection. This can only occur if $L_1=L_j$ and $j \le q \ell _0$ .

If $j> q\ell _0$ , then $n(B)$ is not constant. Consider $n'> n$ such that the intervals $I:=I_n$ , $I':=I_{n'}$ are not empty. Let $J = \psi (I)$ and $J' = \psi (I')$ . We claim that the endpoint B of J cannot be the initial point of $J'$ . Otherwise, $\varphi ^{\circ n}(B)$ would be periodic of period dividing $n'-n$ , contradicting Lemma 5.12. It follows that $\psi (L_j)$ omits intervals of $L_1$ and, therefore, $L_j < L_1$ .

The dynamical space derivative at $\infty $ is controlled by the diameters of $X_j$ .

Lemma 5.14. Assume that $\beta _0$ is a parameter in an order q limb such that $\unicode{x3c9} $ is periodic of period p under $\varphi _{\beta _0}$ and $\{X_0,\ldots , X_{q'-1}\}$ is its cycle of periodic closed balls such that $\unicode{x3c9} \in X_0$ . Then, for all $j \le p$ ,

$$ \begin{align*}\| \partial_z \varphi_{\beta_0}^{\circ j-1}(\infty) \| = \dfrac{\mathrm{diam}(X_j)}{\mathrm{diam}(X_1)}.\end{align*} $$

5.8 Polynomial parameter rescaling

Consider a parameter $\beta _0$ such that $\unicode{x3c9} $ has period p and a non-central return. Recall that $X_0, \ldots , X_{q'-1}$ denotes the periodic closed balls under $\varphi _{\beta _0}$ where $\unicode{x3c9} \in X_0$ . Our analysis will show that there exists a parameter space ball $\Lambda _0$ , around $\beta _0$ , such that $\{X_0, \ldots , X_{q'-1}\}$ is a periodic orbit of closed balls for all $\beta \in \Lambda _0$ . We say that parameters $\beta $ with such an orbit of periodic closed balls, have a period $q'$ primitive renormalization.