Zalcman functions and similarity between the Mandelbrot set, Julia sets, and the tricorn

Lemma 1. Suppose that $c_0 \in \mathbb{M}$ is a Misiurewicz parameter as above. For $k \in \mathbb{N}$ , set $\rho_k := 1/(f_{c_0}^{l+kp})'(c_0)$ . Then we have the following. - (1) The function $\phi_k(w) = f_{c_0}^{l+kp}(c_0 + \rho_k w)$ converges to a non-constant entire function $\phi: \mathbb{C} \to \mathbb{C}$ as $k \to \infty$ uniformly on any compact sets. - (2) There exists a constant $Q \neq 0$ such that the function $$\Phi_k(w) := f_{c_0 + Q\rho_k w}^{l+kp}(c_0 + Q\rho_k w)$$ converges to the same function $\phi(w)$ as $k \to \infty$ uniformly on compact sets of $\mathbb{C}$ . Proof. It is well-known that the sequence of (polynomial) functions $$w \longmapsto f_{c_0}^{kp} \left( a_0 + \frac{w}{\lambda_0^k} \right) \qquad (k \in \mathbb{N})$$ converges to a non-constant entire function $\phi(w)$ uniformly on compact sets of $\mathbb{C}$ . (See Theorem 12 in Appendix. Such a $\phi$ is called a Poincaré function. Indeed, $\phi$ satisfies the functional equation $\phi(\lambda_0 w) = f_{c_0}^p \circ \phi(w)$ , but we will not use it.) Note that this function satisfies $\phi(0) = a_0$ and $\phi'(0) = 1$ . Now let us show (1): set $A_0 := (f_{c_0}^l)'(c_0)$ , where $A_0 \neq 0$ since otherwise $c_0$ is strictly periodic. We also have $(f_{c_0}^{l+kp})'(c_0) = A_0\lambda_0^k = 1/\rho_k$ . For sufficiently small $t \in \mathbb{C}$ , we have the expansion $$f_{c_0}^l(c_0+t) = a_0 + A_0 \cdot t + o(t).$$ Fix an arbitrarily large compact set $E \subset \mathbb{C}$ and take any $w \in E$ . Then by setting $t = w/(A_0\lambda_0^k)$ , $$f_{c_0}^{kp}\left(a_0 + \frac{w}{\lambda_0^k}\right) \sim f_{c_0}^{l+kp}\left(c_0 + \frac{w}{A_0\lambda_0^k} + o(\lambda_0^{-k})\right) \sim f_{c_0}^{l+kp}(c_0 + \rho_k w) = \phi_k(w)$$ when $k \to \infty$ . (Here by $A_k(w) \sim B_k(w)$ we mean $A_k(w) - B_k(w) \to 0$ uniformly on E as $k \to \infty$ .) Hence we have $\phi(w) = \lim_{k \to 0} \phi_k(w)$ on any compact sets. Note that $\phi$ has no poles, since each $\phi_k$ is entire. Next we show (2): suppose that $Q \in \mathbb{C}^*$ is a constant and set $c = c(w) := c_0 + Q\rho_k w$ . We also set $\Phi_k(w) := f_c^{l+kp}(c)$ and $b(c) := f_c^l(c)$ . Recall that a(c) denotes a repelling periodic point (of $f_c$ ) of period p with $a(c_0) = a_0$ , and $\lambda(c)$ denotes its multiplier. Then Theorem 12 in Appendix again implies that the sequence of functions $\phi_k^c(w) := f_c^{kp}(a(c) + w/\lambda(c)^k)$ converges to an entire function $\phi^c(w)$ uniformly on compact sets. In particular, by the proof of Theorem 12, it is not difficult to check that the function $c \mapsto \phi^c(w)$ is holomorphic near $c = c_0$ when we fix a $w \in \mathbb{C}$ . As in Tan's original proof, we employ a theorem on transversality by Douady and Hubbard [DH2, Lemma 1, p.333]: There exists a $B_0 \neq 0$ such that $$b(c) - a(c) = B_0(c - c_0) + o(c - c_0).$$ Hence for $c = c_0 + Q\rho_k w$ (taking w in a compact set), we have $$b(c) = a(c) + B_0 Q \rho_k w + o(\rho_k) = a(c) + \frac{B_0 Q}{A_0} \cdot \frac{\lambda(c)^k}{\lambda_0^k} \cdot \frac{w}{\lambda(c)^k} + o(\rho_k).$$ Set $Q := A_0/B_0$ . Since $\lambda(c)$ is a holomorphic function of c and thus $\lambda(c) = \lambda_0 + O(c - c_0)$ , we have $|\lambda(c)/\lambda_0 - 1| = O(c - c_0)$ . This implies that $$\log \frac{\lambda(c)^k}{\lambda_0^k} = k \cdot O(c - c_0) = O\left(\frac{k}{\lambda_0^k}\right) \to 0 \quad (k \to \infty)$$ for $c = c_0 + Q\rho_k w = c_0 + O(\lambda_0^{-k})$ . Since $$\Phi_k(w) = f_c^{kp}(b(c)) = f_c^{kp}\left(a(c) + \frac{w}{\lambda(c)^k} + o(\rho_k)\right)$$ and $\phi^c(w) \to \phi(w)$ as $c \to c_0$ (uniformly for w in a compact set), we conclude that $$\lim_{k \to \infty} \Phi_k(w) = \phi(w),$$ where the convergence is uniform on any compact sets. Remark. Lemma 1 implies that $c_0 \in J_{c_0} = \partial K_{c_0}$ and $c_0 \in \partial \mathbb{M}$ . Indeed, we can find a $w \in \mathbb{C}$ such that $|\phi(w)| > 2$ and hence $|\phi_k(w)| > 2$ for sufficiently large k. Equivalently, we have $c_0 + \rho_k w \notin K_{c_0}$ for sufficiently large k, where $c_0 + \rho_k w$ tends to $c_0$ as $k \to \infty$ . Since $c_0 \in K_{c_0}$ by definition, we have $c_0 \in J_{c_0}$ . The proof for $c_0 \in \partial \mathbb{M}$ is analogous. The Hausdorff topology. Let us briefly recall the Hausdorff topology of the set of non-empty compact sets $\mathrm{Comp}^(\mathbb{C})$ of $\mathbb{C}$ . For a sequence $\{K_k\}_{k\in\mathbb{N}}\subset\mathrm{Comp}^(\mathbb{C})$ , we say $K_k$ converges to $K\in\mathrm{Comp}^*(\mathbb{C})$ as $k\to\infty$ if for any $\epsilon>0$ , there exists $k_0\in\mathbb{N}$ such that $K\subset\mathrm{N}_{\epsilon}(K_k)$ and $K_k\subset\mathrm{N}_{\epsilon}(K)$ for any $k\geq k_0$ , where $\mathrm{N}_{\epsilon}(\cdot)$ is the open $\epsilon$ neighborhood in $\mathbb{C}$ . Let $\mathbb{D}(r) := \{z \in \mathbb{C} : |z| < r\}$ . For a compact set K in $\mathbb{C}$ , let $[K]_r$ denote the set $(K \cap \mathbb{D}(r)) \cup \partial \mathbb{D}(r)$ . For $a \in \mathbb{C}^*$ and $b \in \mathbb{C}$ , let a(K - b) denote the set of a(z - b) with $z \in K$ . Similarity. Let $c_0$ be a Misiurewicz parameter. Now we state our version of Tan's similarity theorem. Theorem 2 (Similarity between $\mathbb{M}$ and J) There exist a non-constant entire function $\phi$ on $\mathbb{C}$ , a sequence $\rho_k \to 0$ , and a constant $q \neq 0$ such that if we set $\mathcal{J} := \phi^{-1}(J_{c_0}) \subset \mathbb{C}$ , then for any large constant r > 0, we have (a) $$\left[\rho_k^{-1}(J_{c_0}-c_0)\right]_r \to [\mathcal{J}]_r$$ , and (b) $$\left[\rho_k^{-1}q(\mathbb{M}-c_0)\right]_r \to [\mathcal{J}]_r$$ as $k \to \infty$ in the Hausdorff topology. Proof of (a). Let $\phi_k$ , $\phi$ , and $\rho_k = 1/(A_0\lambda_0^k)$ be as given in the proof of Lemma 1. Since $f_{c_0}^n(J_{c_0}) = J_{c_0}$ , we have $\left[\rho_k^{-1}(J_{c_0} - c_0)\right]_r = \left[\phi_k^{-1}(J_{c_0})\right]_r$ . By $[\mathcal{J}]_r = [\phi^{-1}(J_{c_0})]_r$ and uniform convergence of $\phi_k \to \phi$ on $\overline{\mathbb{D}(r)}$ , the claim easily follows. Proof of (b). Let q := 1/Q (where $Q \neq 0$ is defined in the proof of Lemma 1) and $\mathcal{M}_k := \rho_k^{-1} q(\mathbb{M} - c_0)$ . Fix any $\epsilon > 0$ . Since the set $\overline{\mathbb{D}(r)} - N_{\epsilon}(\mathcal{J})$ is compact, there exists an $N = N(\epsilon)$ such that $|f_{c_0}^N \circ \phi(w)| > 2$ for any $w \in \overline{\mathbb{D}(r)} - N_{\epsilon}(\mathcal{J})$ . By uniform convergence of $\Phi_k(w) = f_{c_0+Q\rho_k w}^{l+kp}(c_0 + Q\rho_k w)$ to $\phi(w)$ on compact sets in $\mathbb{C}$ (Lemma 1), we have $$|f_{c_0+Q\rho_k w}^{(l+kp)+N}(c_0+Q\rho_k w)| > 2$$ for all sufficiently large k. This implies that $c_0 + Q\rho_k w \notin \mathbb{M}$ , equivalently, $w \notin \mathcal{M}_k$ . Hence we have $$[\mathcal{M}_k]_r \subset \mathrm{N}_{\epsilon}([\mathcal{J}]_r).$$ Next we show the opposite inclusion $[\mathcal{J}]_r \subset N_{\epsilon}([\mathcal{M}_k]_r)$ for k large enough. Let us approximate $[\mathcal{J}]_r$ by a finite subset E of $[\mathcal{J}]_r$ such that the $\epsilon/2$ neighborhood of E covers $[\mathcal{J}]_r$ . Now it is enough to prove that for any $w_0 \in E$ , there exists a sequence $w_k \in [\mathcal{M}_k]_r$ such that $|w_0 - w_k| < \epsilon/2$ for sufficiently large k. Let $\Delta$ be a disk of radius $\epsilon/2$ centered at $w_0$ . When $\Delta \cap \partial \mathbb{D}(r) \neq \emptyset$ , we can take such a $w_k$ in $\partial \mathbb{D}(r)$ . Hence we may assume that $\Delta \subset \mathbb{D}(r)$ . Since $\phi(w_0) \in J_{c_0}$ and repelling cycles are dense in $J_{c_0}$ (see [Sch] and [Mi2]. See also the remark below), we can choose a $w'_0$ such that $\phi(w'_0)$ is a repelling periodic point of some period m and $|w_0 - w'_0| < \epsilon/4$ . This implies that the function $\chi : w \mapsto f_{c_0}^m(\phi(w)) - \phi(w)$ has a zero at $w = w'_0$ . Let us consider the function $\chi_k : w \mapsto f_{c_0+Q\rho_k w}^m(\Phi_k(w)) - \Phi_k(w)$ , where $\Phi_k(w) = f_{c_0+Q\rho_k w}^{l+kp}(c_0+Q\rho_k w)$ as in Lemma 1. By the Hurwitz theorem and uniform convergence of $\Phi_k$ to $\phi$ on compact sets of $\mathbb{C}$ , $\chi_k$ has a zero $w_k$ in $\Delta$ and $|w_k - w_0'| < \epsilon/4$ for all sufficiently large k. In particular, $c_k := c_0 + Q\rho_k w_k$ satisfies $f_{c_k}^{m+(l+kp)}(c_k) = f_{c_k}^{l+kp}(c_k)$ and thus $c_k \in \mathbb{M}$ . Hence we have a desired $w_k \in \mathcal{M}_k$ with $|w_k - w_0| < \epsilon/2$ . Example (Calculation of Q). When $c_0 = -2$ (hence l = p = 1), $A_0 = f'_{c_0}(-2) = -4$ and $\lambda_0 = f'_{c_0}(f_{c_0}(-2)) = 4$ . Since $a(c)^2 + c = a(c)$ we find da(c)/dc = -1/(2a(c) - 1). Moreover, $db(c)/dc = (d/dc)(c^2 + c) = 2c + 1$ . Hence for $c = c_0 = -2$ , we have $B_0 = -3 - (-1/3) = -8/3$ and the constant Q is $A_0/B_0 = 3/2$ .

Proposition 3. (J and $\partial \mathbb{M}$ as non-normality loci) For the quadratic family $f_c(z) = z^2 + c$ ( $c \in \mathbb{C}$ ), the Julia sets and the boundary of the Mandelbrot set are characterized as follows: - (1) For each $c \in \mathbb{C}$ , the Julia set $J_c = \partial K_c$ is the set of points where the family $\{z \mapsto f_c^n(z)\}_{n \geq 0}$ is not normal. - (2) The boundary $\partial \mathbb{M}$ of the Mandelbrot set is the set of points where the family $\{c \mapsto f_c^n(c)\}_{n \geq 0}$ is not normal. The proof is done by an elementary equicontinuity argument. For the Mandelbrot set case, see [Mc, Theorem 4.6]. Dynamical Zalcman functions. Let $\mathcal{F}_c$ denote the family $\{z \mapsto f_c^n(z)\}_{n\geq 0}$ of polynomial functions on $\mathbb{C}$ for each parameter $c \in \mathbb{C}$ . By Proposition 3, we can apply Zalcman's lemma to this family and obtain the sets $$\mathcal{Z}_c := \mathcal{Z}(\mathcal{F}_c)$$ and $\mathcal{Z}_c(z_0) := \mathcal{Z}(\mathcal{F}_c, z_0)$ of dynamical Zalcman functions of $f_c$ (for each $z_0 \in J_c$ ). These sets have a good invariance with respect to the operations ' $f_c$ o' and 'oAff', as shown by Steinmetz [St, Theorems 1 and 2]: Proposition 4 (Invariance) For each $z_0 \in J_c$ , the family $\mathcal{Z}_c(z_0)$ satisfies $$f_c \circ \mathcal{Z}_c(z_0) = \mathcal{Z}_c(z_0) = \mathcal{Z}_c(z_0) \circ \text{Aff.}$$ More precisely, - (1) If $\phi \in \mathcal{Z}_c(z_0)$ then $f_c \circ \phi \in \mathcal{Z}_c(z_0)$ and $\phi = f_c \circ \phi_1$ for some $\phi_1 \in \mathcal{Z}_c(z_0)$ . - (2) For any $A \in \text{Aff}$ and $\phi \in \mathcal{Z}_c(z_0)$ , we have $\phi \circ A^{\pm 1} \in \mathcal{Z}_c(z_0)$ . Note that the universal space $\mathcal{U}$ only satisfies $f_c \circ \mathcal{U} \subset \mathcal{U}$ and $\mathcal{U} \circ \mathrm{Aff} = \mathcal{U}$ . Proposition 4 implies that we also have $f_c \circ \mathcal{Z}_c = \mathcal{Z}_c = \mathcal{Z}_c \circ \text{Aff}$ . However, the equality $\mathcal{Z}_c = \mathcal{Z}_c(z_0)$ holds for any $z_0 \in J_c$ in most cases. To see this, we introduce some terminology: The univalent grand orbit $UGO(z_0)$ of $z_0 \in \mathbb{C}$ is the set of $\zeta$ such that $f_c^m(z_0) = f_c^n(\zeta)$ for some $m, n \in \mathbb{N}$ and there is a univalent branch g of $f_c^{-n} \circ f_c^m$ in a neighborhood of $z_0$ with $g(z_0) = \zeta$ . The postcritical set $P_c$ of $f_c$ is the closure of the orbit $\{c, f_c(c), f_c^2(c), \cdots\}$ . (Note that c is a unique critical value of $f_c$ .) We say $f_c$ satisfies (\*)-condition if (\*): For any $z_0 \in J_c$ , there exists a $\zeta \in UGO(z_0) - P_c$ . Now we claim the following:

Theorem 5. If $f_c$ satisfies (\*)-condition, then $\mathcal{Z}_c = \mathcal{Z}_c(z_0)$ for any $z_0 \in J_c$ . Moreover, the set of such c contains $\mathbb{C} - \partial \mathbb{M}$ , which is an open and dense subset of $\mathbb{C}$ , and the Misiurewicz parameters in $\partial \mathbb{M}$ except $c_0 = -2$ . Proof. The first claim of the theorem is an immediate corollary of [Ka, Théorème 3.8]. For the second claim, note that $\partial \mathbb{M}$ consists of the parameters c for which $f_c$ is expanding (hyperbolic) or infinitely renormalizable (see [Mc, §4 and §8]). Since expanding $f_c$ does not contain any critical point in the Julia set $J_c$ , it satisfies (\)-condition. Any infinitely renormalizable $f_c$ satisfies (\)-condition by [Ka, Proposition 3.7]. Now suppose that c is a Misiurewicz parameter, satisfying $f_c^l(c) = f_c^{l+p}(c)$ with minimal $l, p \ge 1$ . Then $z_0 \in P_c$ if and only if $z_0 = f_c^k(c)$ for some $0 \le k \le l+p-1$ . For such a $z_0 \in P_c$ , when $l \ge 2$ or $p \ge 2$ , one can easily find some $\zeta \in UGO(z_0) - P_c$ . Otherwise l = p = 1, equivalent to c = -2. In this case we have $P_{-2} = \{-2, 2\} = UGO(-2) = UGO(2)$ . If $z_0 \notin P_c$ , $z_0$ itself is an element of $UGO(z_0) - P_c$ . Hence $f_c$ for the Misiurewicz parameter c satisfies (\*)-condition unless c = -2. Remark. When $f_c$ satisfies (\*)-condition, the dynamical Zalcman function $\mathcal{Z}_c$ can be an alternative ingredient for the Lyubich-Minsky Riemann surface lamination for $f_c$ . See [Ka, §3] for more details, and see [LM] for Lyubich and Minsky's lamination theory in complex dynamics. Parametric Zalcman functions. Let $\mathcal{Q}$ denote the family $\{c \mapsto f_c^n(c)\}_{n \geq 0}$ of polynomial functions on $\mathbb{C}$ . Again by Proposition 3, we can apply Zalcman's lemma to this family and obtain the sets $$\mathcal{P} := \mathcal{Z}(\mathcal{Q})$$ and $\mathcal{P}(c_0) := \mathcal{Z}(\mathcal{Q}, c_0)$ of parametric Zalcman functions of the quadratic family $\{f_c(z) = z^2 + c\}_{c \in \mathbb{C}}$ (for each $c_0 \in \partial \mathbb{M}$ ). These sets have weaker invariance, which is also pointed out by Steinmetz [St, Remark in §1]: Proposition 6 (Invariance for $\mathcal{P}$ ) For each $c_0 \in \partial \mathbb{M}$ , the family $\mathcal{P}(c_0)$ satisfies $$f_{c_0} \circ \mathcal{P}(c_0) = \mathcal{P}(c_0) = \mathcal{P}(c_0) \circ \text{Aff.}$$ Hence we only have $\mathcal{P} = \mathcal{P} \circ \text{Aff}$ for the total space $\mathcal{P}$ of the parametric Zalcman functions. In a forthcoming paper we will present a general account on dynamical and parametric Zalcman functions for families of rational functions parametrized by Riemann surfaces. Dynamical-parametric intersection and similarity. By Proposition 4 and Proposition 6 above, if $c_0 \in J_{c_0}$ and $c_0 \in \partial \mathbb{M}$ , then $\mathcal{Z}_{c_0}(c_0)$ and $\mathcal{P}(c_0)$ exhibit the same invariance in the universal space $\mathcal{U}$ . Hence one might expect that there exists some $\phi \in \mathcal{Z}_{c_0}(c_0) \cap \mathcal{P}(c_0)$ when $c_0 \in J_{c_0} \cap \partial \mathbb{M}$ . Indeed, the existence of such an intersection implies asymptotic similarity between $J_{c_0}$ and $\mathbb{M}$ at $c_0$ : Theorem 7 (Intersection implies similarity) Suppose that $\mathcal{Z}_{c_0}(c_0) \cap \mathcal{P}(c_0) \neq \emptyset$ for some $c_0 \in \partial \mathbb{M}$ . More precisely, there exist sequences of affine maps $A_k$ , $B_k \in \text{Aff}$ and positive integers $m_k$ , $n_k \in \mathbb{N}$ such that as $k \to \infty$ we have - (1) Both $A_k$ and $B_k$ converge to the same constant map $c_0$ in $\overline{\mathcal{U}}$ ; and - (2) Both $f_{c_0}^{n_k}(A_k(w))$ and $f_{B_k(w)}^{n_k}(B_k(w))$ converge to the same entire function $\phi(w)$ in $\mathcal{U}$ . Suppose in addition that $c_0 \in J_{c_0}$ and let $\mathcal{J} := \phi^{-1}(J_{c_0}) \subset \mathbb{C}$ . Then for any large constant r > 0, we have (a) $$\left[A_k^{-1}(J_{c_0})\right]_r \to [\mathcal{J}]_r$$ , and (b) $$\left[B_k^{-1}(\mathbb{M})\right]_r \to [\mathcal{J}]_r$$ as $k \to \infty$ in the Hausdorff topology. Proof. Condition (1) implies that if we write $A_k(w) = c_k + \rho_k w$ and $B_k(w) = c'_k + \rho'_k w$ then $c_k, c'_k \to c_0$ and $\rho_k, \rho'_k \to 0$ as $k \to \infty$ . Then the proof is only a slight modification of that of Theorem 2. Intersection at Misiurewicz parameters. Suppose that $c_0$ is a Misiurewicz parameter. (Hence $c_0 \in J_{c_0}$ and $c_0 \in \partial \mathbb{M}$ by the remark below the proof of Lemma 1.) By the construction of the entire function $\phi$ in Lemma 1, we obviously have $\phi \in \mathcal{Z}_{c_0}(c_0)$ and $\phi \in \mathcal{P}(c_0)$ . Thus we conclude the following: Theorem 8 (Intersection) For any Misiurewicz parameter $c_0$ , $\mathcal{Z}_{c_0}(c_0)$ and $\mathcal{P}(c_0)$ share at least one element $\phi \in \mathcal{U}$ . Note that the set of Misiurewicz parameters is a dense subset of $\partial M$ . (This can be shown by a standard normal family argument. See Levin [Le] or [Ka, Théorème 1.1 (1)].) In [Ka] the author proved Lemma 1 for a wider class of parameters in $\partial \mathbb{M}$ , called semi-hyperbolic parameters. Shishikura proved in [Sh] that the set of semi-hyperbolic parameters is a dense subset of $\partial \mathbb{M}$ of Hausdorff dimension 2. Hence we also have a generalization of Theorem 8 for semi-hyperbolic parameters. Question. Besides the example of Lemma 1 and its generalization to semi-hyperbolic parameters, are there any other intersections of the sets of dynamical and parametric Zalcman functions?

Lemma 10. Suppose that $c_0 \in \mathbb{T}$ is a Misiurewicz parameter as above. For $k \in \mathbb{N}$ , set $\rho_k := 1/(g_{c_0}^{2l+2kp})'(c_0)$ . Then we have the following. - (1) The function $\phi_k(w) = g_{c_0}^{2l+2kp}(c_0 + \rho_k w)$ converges to a non-constant entire function $\phi: \mathbb{C} \to \mathbb{C}$ as $k \to \infty$ uniformly on any compact sets. - (2) There exists a real linear transformation $H(w) = Qw + Q'\overline{w}$ with $|Q| \neq |Q'|$ such that the function $$\Phi_k(w) := g_{c_0 + H(\rho_k w)}^{2l + 2kp} (c_0 + H(\rho_k w))$$ converges to the same function $\phi(w)$ as $k \to \infty$ uniformly on compact sets of $\mathbb{C}$ .  Figure 2: (T) The tricorn T. (JT1) Center: −1.2222454262925588 + 0.18411010266019595i, square width: from 6.0 to 0.005. (JT2) Center: −1.0672232757314006 + 0.13470887783195631i, square width: from 6.0 to 0.001.  Figure 3: (MJT) Center: -1.4303576324513074, square width: from 7.0 to 0.005. Note that $\Phi_k(w)$ is a real analytic function of w but it converges to an entire function $\phi(w)$ . We prove (1) and (2) separately. Proof of (1). Both $g_{c_0}^{2l}$ and $g_{c_0}^{2p}$ are holomorphic and thus the usual derivatives $A_0 := (g_{c_0}^{2l})'(c_0) \neq 0$ and $\lambda_0 = (g_{c_0}^{2p})'(a_0)$ make sense. Since $a_0 = g_{c_0}^{2l}(c_0)$ is a repelling fixed point (that is, $|\lambda_0| > 1$ ) of $g_{c_0}^{2p}$ , the Poincaré function $$\phi(w) := \lim_{n \to \infty} g_{c_0}^{2kp} \left( a_0 + \frac{w}{\lambda_0^k} \right)$$ exists and is an entire function, where the convergence is uniform on compact sets. (See Theorem 12 in Appendix below.) By the expansion $g_{c_0}^{2l}(c_0+t)=a_0+A_0t+o(t)$ ( $t\to 0$ ) we obtain $$\phi(w) = \lim_{n \to \infty} g_{c_0}^{2l+2kp} \left( c_0 + \frac{w}{A_0 \lambda_0^k} \right).$$ Hence we set $\rho_k := (A_0 \lambda_0^k)^{-1} = 1/(g_{c_0}^{2l+2kp})'(c_0)$ . To show (2), we need an extra lemma about stability and transversality of repelling periodic point $a_0$ :

Theorem 12. Let $g: \mathbb{C} \to \mathbb{C}$ be an entire function with g(0) = 0, $g'(0) = \lambda$ , and $|\lambda| > 1$ . Then the sequence $\phi_n(w) = g^n(w/\lambda^n)$ converges uniformly on compact sets in $\mathbb{C}$ . Moreover, the limit function $\phi: \mathbb{C} \to \mathbb{C}$ satisfies $g \circ \phi(w) = \phi(\lambda w)$ and $\phi'(0) = 1$ . Proof. Since $g(z) = \lambda z + O(z^2)$ near z = 0, there exists a disk $\Delta = \mathbb{D}(\delta) = \{z \in \mathbb{C} : |z| < \delta\}$ such that $g|\Delta$ is univalent and $\Delta \in g(\Delta)$ . Hence we have a univalent branch $g_0^{-1}$ of g that maps $\Delta$ into itself. First we show that $\phi_n$ is univalent on $\mathbb{D}(\delta/4)$ : Since the map $\phi_n^{-1}: w \mapsto \lambda^n g_0^{-n}(w)$ is well-defined on $\Delta = \mathbb{D}(\delta)$ and univalent, its image contains $\mathbb{D}(\delta/4)$ by the Koebe 1/4 theorem. Hence $\phi_n$ is univalent on $\mathbb{D}(\delta/4)$ , and by the Koebe distortion theorem, the family $\{\phi_n\}_{n\geq 0}$ is locally uniformly bounded on $\mathbb{D}(\delta/4)$ and thus equicontinuous. Next we show that $\phi_n$ has a limit on $\mathbb{D}(\delta/4)$ : Fix an arbitrarily large r > 0 and an integer N such that $r < \delta|\lambda|^N/4$ . By using the Koebe 1/4 theorem as above, the function $G_{N,k}(w) := \lambda^N g^k(w/\lambda^{N+k})$ $(k \in \mathbb{N})$ satisfying $\phi_{N+k} = \phi_N \circ G_{N,k}$ is univalent on the disk $\mathbb{D}(\delta|\lambda|^N/4)$ . By the Koebe distortion theorem, there exists a constant C > 0 independent of N and k such that for any $w \in \mathbb{D}(r)$ and sufficiently large N we have $|G'_{N,k}(w) - 1| \le C|w|/|\lambda|^N$ . By integration we have $|G_{N,k}(w) - w| \le Cr^2/(2|\lambda|^N)$ on $\mathbb{D}(r)$ . In particular, $G_{N,k} \to \mathrm{id}$ uniformly on $\mathbb{D}(\delta/4)$ as $N \to \infty$ . Since the family $\{\phi_n\}$ is equicontinuous on $\mathbb{D}(\delta/4)$ , the relation $\phi_{N+k} = \phi_N \circ G_{N,k}$ implies that $\{\phi_n\}_{n \ge 0}$ is Cauchy and has a unique limit $\phi$ on any compact sets in $\mathbb{D}(\delta/4)$ . Let us check that the convergence extends to $\mathbb{C}$ : (We will not use the functional equation $g^n \circ \phi(w) = \phi(\lambda^n w)$ . Compare [Mi2, Cor.8.12].) Since $|\phi_{N+k}(w) - \phi_N(w)| = |\phi_N(G_{N,k}(w)) - \phi_N(w)|$ and $|G_{N,k}(w) - w| = Cr^2/(2|\lambda|^N)$ on $\mathbb{D}(r)$ , it follows that the family $\{\phi_{N+k}\}_{k\geq 0}$ (with fixed N) is uniformly bounded on $\mathbb{D}(r)$ . Hence $\{\phi_n\}_{n\geq 0}$ is normal on any compact set in $\mathbb{C}$ and any sequential limit coincides with the local limit $\phi$ on $\mathbb{D}(\delta/4)$ . The equation $g \circ \phi(w) = \phi(\lambda w)$ and $\phi'(0) = 1$ are immediate from $g \circ \phi_n(w) = \phi_{n+1}(\lambda w)$ and $\phi'_n(0) = 1$ . Remark. One can easily extend this proof to the case of meromorphic g by using the spherical metric. Acknowledgement. The author would like to thank Hiroyuki Inou for suggesting the bi-quadratic family for the proof of Lemma 11. This work is partly supported by JSPS KAKENHI Grants Number 16K05193 and Number 19K03535.