Abstract

For the quadratic family $f_c\left(z\right)=z^2+c$ , we study the exterior Green function of the Mandelbrot set $ℳ$ through classical conformal uniformization, logarithmic potential theory, and quantitative escape-rate approximation. As classical input, we use the Douady–Hubbard parameter Böttcher map ${\Phi }_{ℳ}:ℂ\backslash ℳ\to ℂ\backslash \overline{\mathbb{D}}$ , normalized at infinity, and the representation $G_{ℳ}\left(c\right)=\log \left|{\Phi }_{ℳ}\left(c\right)\right|$ . The harmonicity and positivity of $G_{ℳ}$ on $ℂ\backslash ℳ$ , its continuous extension by zero on $ℳ$ , its normalized logarithmic growth at infinity, its uniqueness as the normalized exterior Green function, and the identity $G_{ℳ}\left(c\right)=g_c\left(c\right)=2g_c\left(0\right)$ are classical facts recalled to fix the analytic framework and normalization. The quantitative escape-rate estimate used in this paper was established in the author’s earlier work. The contribution of the present paper is to provide a self-contained telescoping derivation and to identify the parameter-plane conformal meaning of that estimate through the preceding classical identity. Through this interpretation, the previously established $O\left(2^{-n}\right)$ escape-rate estimate becomes an explicit quantitative approximation of the parameter Böttcher potential by the normalized finite critical orbit approximants. We further obtain uniform exponential convergence on compact subsets of $ℂ\backslash ℳ$ , prove Hausdorff continuity of positive equipotentials over compact level ranges, and formulate fixed-index and variable-time numerical procedures consistent with the analytic normalization. For detected escaping parameters, the variable-time first-escape estimator is accompanied by an explicit pointwise error certificate. The numerical figures are finite-resolution illustrations of the analytic results and do not certify membership in $ℳ$ on the unresolved finite-time mask.

Keywords

Mandelbrot Set, Green Function, Böttcher Coordinate, Exterior Uniformization, Escape-Rate Approximation, Equipotentials

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1. Introduction

The quadratic family $f_c\left(z\right)=z^2+c,c\in ℂ$ introduced formally in Definition 3.1, occupies a central position in one-dimensional holomorphic dynamics. For a fixed parameter $c$ , its dynamical-plane structure is encoded by the filled Julia set $K_c$ , the Julia set $J_c=\partial K_c$ , and the basin of infinity $A_c\left(\infty \right)=ℂ\backslash K_c$ . Its parameter-space organization is governed by the Mandelbrot set

$ℳ=\left\{c\in ℂ:{\left\{f_c^{\circ n}\left(0\right)\right\}}_{n\ge 0}\text{is}\text{bounded}\right\}.$ (1)

The formal definition is given in Definition 3.3. The critical orbit $0,f_c\left(0\right)=c,f_c^{\circ 2}\left(0\right),\cdots$ therefore determines whether the parameter belongs to $ℳ$ or to its complement.

In the classical theory developed by Douady and Hubbard, the exterior $ℂ\backslash ℳ$ admits a normalized conformal uniformization

${\Phi }_{ℳ}:ℂ\backslash ℳ\to ℂ\backslash \overline{\mathbb{D}},\frac{{\Phi }_{ℳ}\left(c\right)}{c}\to 1\left(c\to \infty \right).$(2)

Under the uniformization (2), the associated parameter Green function is represented by

$G_{ℳ}\left(c\right)=\log \left|{\Phi }_{ℳ}\left(c\right)\right|.$ (3)

The representation (3), together with the classical properties of ${\Phi }_{ℳ}$ , yields the harmonicity and strict positivity of $G_{ℳ}$ on $ℂ\backslash ℳ$ , its continuous extension by zero on $ℳ$ , and its normalized logarithmic growth

$G_{ℳ}\left(c\right)-\text{log}\left|c\right|\to 0\left(c\to \infty \right).$ (4)

The boundary condition and the growth condition (4) enter the classical uniqueness characterization of $G_{ℳ}$ as the normalized exterior Green function with pole at infinity. The same conformal coordinate determines the classical parameter equipotentials and external rays. These facts are standard consequences of classical exterior uniformization and logarithmic potential theory. They are recalled here to establish the analytic framework and normalization required for the subsequent quantitative analysis; see [1]-[5]. The relationship between the dynamical-plane and parameter-plane normalizations is likewise classical. Let $g_c$ denote the dynamical Green function of $f_c$ . Its functional equation is

$g_c\left(f_c\left(z\right)\right)=2g_c\left(z\right).$ (5)

Applying (5) at $z=0$ , together with $f_c\left(0\right)=c$ , gives

$G_{ℳ}\left(c\right)=g_c\left(c\right)=2g_c\left(0\right),c\in ℂ\backslash ℳ.$ (6)

Combining (3) and (6) gives the complete normalization identity

$G_{ℳ}\left(c\right)=\log \left|{\Phi }_{ℳ}\left(c\right)\right|=g_c\left(c\right)=2g_c\left(0\right).$ (7)

The classical identity (7) provides the bridge between critical orbit escape and the parameter Böttcher potential. The representation of $G_{ℳ}$ by the parameter Böttcher map, its standard potential-theoretic properties, its uniqueness characterization, and the preceding dynamical–parameter identity are not claimed as new results. They are included to provide a self-contained structural setting and to establish the precise normalization used throughout the paper.

For the critical orbit

$z_0\left(c\right)=0,z_{n+1}\left(c\right)=z_n{\left(c\right)}^2+c.$ (8)

consider the finite escape-rate approximants

$G_n\left(c\right)=\frac{1}{2^n}{\text{log}}^{+}\left|z_n\left(c\right)\right|=\frac{1}{2^n}{\text{log}}^{+}\left|f_c^{\circ n}\left(0\right)\right|.$ (9)

The quantitative escape-rate estimate governing the approximants in (9) was established in the author’s earlier work [6]. Accordingly, the estimate considered in isolation is not claimed as a new result of the present paper.

For completeness, the present manuscript gives a self-contained telescoping derivation directly from the critical orbit recurrence in (8). More importantly, the present manuscript identifies the parameter-plane conformal meaning of the estimate through the classical normalization identity (7). Through the identification (7), the normalized finite critical orbit quantities

$A_n\left(c\right):=2G_n\left(c\right).$ (10)

become quantitative approximations to the parameter Böttcher potential. Once the critical orbit has entered a prescribed escape region, Theorem 8.1 gives an explicit estimate of the form

$\left|2G_n\left(c\right)-\log \left|{\Phi }_{ℳ}\left(c\right)\right|\right|\le C\left(c,R\right)2^{-n}.$ (11)

In (11), the constant $C\left(c,R\right)$ is independent of $n$ . Thus, through the conformal interpretation developed here, the previously established escape-rate estimate becomes an explicit approximation result for the exterior Green function of the Mandelbrot set. The pointwise estimate (11) admits a uniform compact-subset consequence. The present paper also obtains uniform exponential convergence on every compact subset of $ℂ\backslash ℳ$ . More precisely, if $K\subset ℂ\backslash ℳ$ is compact, then Corollary 8.4 provides common constants $n_K\ge 1$ and $C_K>0$ such that

$\underset{c\in K}{\text{sup}}\left|2G_n\left(c\right)-G_{ℳ}\left(c\right)\right|\le C_K{2}^{-n},n\ge n_K.$ (12)

The compact-uniform estimate (12) clarifies why a common finite iteration depth can approximate $G_{ℳ}$ reliably on parameter regions lying a positive distance from $\partial ℳ$ . It also identifies the boundary as the region in which no global common escape time should be expected.

The numerical methodology is formulated to preserve the distinction between fixed-index approximation and parameter-dependent escape detection. For a prescribed index $N$ , the fixed-index field is

$A_N\left(c\right)=2G_N\left(c\right).$ (13)

This quantity uses the same iteration index for every parameter. By contrast, the variable-time first-escape estimator is

${\widehat{G}}_{\text{esc}}\left(c\right)=2G_{{\tau }_{N_{\text{max}}}\left(c\right)}\left(c\right).$(14)

where ${\tau }_{N_{\text{max}}}\left(c\right)$ is the first index, not exceeding the iteration budget $N_{\text{max}}$ , at which the critical orbit crosses the chosen escape radius. The estimators in (13) and (14) coincide only when $N={\tau }_{N_{\text{max}}}\left(c\right)$ . They are therefore kept notationally and computationally distinct. For every parameter whose escape is detected, the first-escape estimator is accompanied by the explicit pointwise certificate

$\left|{\widehat{G}}_{\text{esc}}\left(c\right)-G_{ℳ}\left(c\right)\right|\le {\widehat{E}}_{\text{esc}}\left(c\right).$(15)

The certificate (15) applies only when escape is detected and the stated admissibility condition on the escape radius is satisfied. If escape is not detected within the prescribed iteration budget, the parameter is placed in a finite-time unresolved mask. This numerical outcome records only the failure to detect escape and does not certify membership in $ℳ$ . The corresponding dark regions in the figures must therefore not be interpreted as rigorous numerical representations of the Mandelbrot set.

The convergence visualization similarly distinguishes the exact Green function from a certified numerical reference. If $G_{\text{ref}}\left(c\right)$ is a first-escape reference computed using a large reference radius and

$\left|G_{\text{ref}}\left(c\right)-G_{ℳ}\left(c\right)\right|\le E_{\text{ref}}\left(c\right).$ (16)

then the triangle inequality gives

$\left|A_n\left(c\right)-G_{\text{ref}}\left(c\right)\right|\le B_n\left(c;R\right)+E_{\text{ref}}\left(c\right).$ (17)

Here $B_n\left(c;R\right)$ is the bound supplied by Theorem 8.1. The right-hand side of (17) is the certified comparison bound represented by the dotted curves in the convergence Figure 5.

The exterior conformal representation also gives the identity for every positive equipotential

$\begin{array}{l}{E}_t:=\left\{c\in ℂ\backslash ℳ:G_{ℳ}\left(c\right)=t\right\}\\ ={\Phi }_{ℳ}^{-1}\left(C_t\right),\text{where},C_t:=\left\{w\in ℂ:\left|w\right|={\text{e}}^t\right\}.\end{array}$ (18)

The present paper proves that the family of equipotentials varies continuously in the Hausdorff metric over every compact positive level range. More precisely, Theorem 9.1 shows that if $0<a<b$ , $t_n,t\in \left[a,b\right]$ , and $t_n\to t$ , then

$d_{\text{H}}\left(E_{t_n},E_t\right)\to 0.$ (19)

The convergence in (19) concerns the exact equipotentials of $G_{ℳ}$ . The numerical level curves displayed later are finite-resolution illustrations of the theorem and are not used in its proof.

The computed exterior-geometry figure also uses a truncated approximation to the Böttcher logarithm in order to visualize numerical external-angle fields and parameter-ray segments. These objects are included only as finite-resolution approximations. No global branch-certified error estimate for $\arg {\Phi }_{ℳ}$ is claimed, and the extracted numerical ray components are not identified with exact external rays.

The contributions developed in the present paper may therefore be summarized as follows:

(i) A self-contained telescoping derivation of the previously established quantitative escape-rate estimate;

(ii) An explicit interpretation of that estimate as an approximation to the parameter Böttcher potential $\log \left|{\Phi }_{ℳ}\left(c\right)\right|$ ;

(iii) Uniform exponential convergence of the normalized finite critical orbit approximants on compact subsets of $ℂ\backslash ℳ$ ;

(iv) A precise distinction between fixed-index approximation and variable-time first-escape estimation, including a pointwise error certificate for detected escaping parameters;

(v) A certified comparison bound for the discrepancy between fixed-index approximants and the numerical first-escape reference;

(vi) Hausdorff continuity of the exact equipotentials over compact positive level ranges;

(vii) Finite-resolution numerical procedures whose notation and normalization are consistent with the analytic quantities being approximated.

The emphasis of the present article therefore differs from that of the author’s earlier work [6]. The earlier paper developed the effective quantitative behaviour of regularized critical orbit escape. The present paper places that estimate within the classical exterior conformal geometry of the Mandelbrot set, supplies the telescoping derivation for self-containment, derives the compact-uniform consequence, formulates the certified first-escape comparison, and establishes the positive-level equipotential-stability result. Accordingly, the contributions of the present article lie in the quantitative–conformal interpretation, its compact-uniform consequence, the certified formulation of the numerical approximations, and the Hausdorff stability analysis. The classical construction of the parameter Green function, its standard potential-theoretic properties, its uniqueness characterization, the dynamical–parameter normalization, and the escape-rate estimate considered independently are not claimed as new results.

The paper is organized as follows. Section 2 reviews the relevant literature and places the present work in its mathematical context. Section 3 introduces the definitions and normalizations used throughout the paper. Section 4 recalls the classical dynamical Böttcher coordinate, its holomorphic dependence near infinity, and the Douady–Hubbard parameter uniformization, summarized schematically in Figure 1. Section 5 records the classical exterior conformal representation of the parameter Green function and derives its standard potential-theoretic properties. Section 6 gives, for completeness, a direct maximum-principle proof of the classical uniqueness characterization. Section 7 recalls the exact conformal description of equipotentials and external rays. Section 8 gives the self-contained telescoping derivation, establishes the quantitative–conformal interpretation, derives uniform exponential convergence on compact exterior parameter sets, and introduces the numerical procedures used in Figures 2-5. Section 9 proves Hausdorff continuity of equipotentials over compact positive level ranges, illustrated in Figure 6. Section 10 explains the relation between the present conformal interpretation and the author’s earlier quantitative study. Section 11 records the limitations of the present analysis, and Section 12 formulates the proposed directions for future research. The paper concludes in Section 13.

2. Literature Review and Mathematical Context

The modern study of quadratic complex dynamics is rooted in the work of Douady and Hubbard, who developed the foundational theory of polynomial-like mappings, external uniformization, and the parameter-space description of the Mandelbrot set [2] [3]. In their framework, the Mandelbrot set appears as the connectedness locus for the quadratic family, while its complement is naturally described by a Green function with pole at infinity. Earlier potential-theoretic foundations of polynomial iteration, including invariant sets and capacity, go back to Brolin’s work [7].

A standard and geometrically transparent account of these ideas is given by Milnor [5], who emphasizes the role of critical orbits, Julia sets, Böttcher coordinates, and external rays in one-dimensional holomorphic dynamics. Closely related presentations appear in Carleson and Gamelin [1] and in Beardon’s classical treatment of iteration theory [8]. For broader analytic and geometric background, including Teichmüller-theoretic aspects and the global organization of parameter space, one may also consult Hubbard [4].

From the standpoint of analysis, the Green function of $ℂ\backslash ℳ$ belongs to classical logarithmic potential theory. Its basic properties—harmonicity on the complement, vanishing on the compact set, and logarithmic growth at infinity—fit naturally into the theory of harmonic and subharmonic functions, planar Green functions, Dirichlet problems, and conformal mapping. Useful background for these topics may be found in Ransford [9] and in Axler, Bourdon, and Ramey [10]. In parameter spaces of rational maps, potential-theoretic and Green-function ideas also underlie bifurcation currents and Lyapunov-exponent methods, as developed for example by DeMarco [11] and Dujardin–Favre [12]. For Green functions in varying polynomial families, see also Favre–Gauthier [13].

Recent developments show that the geometry of the Mandelbrot set and of related parameter spaces continues to evolve in several active directions. These include renormalization and local-connectivity phenomena associated with the Mandelbrot set and the MLC programme [14], self-similarity near Siegel parameters and pacman renormalization [15], positive-area phenomena for quadratic Julia sets [16], continuity and limiting behaviour for Julia sets and invariant external rays [17], richer parameter-space structures in cubic and parabolic settings [18], and Mandelbrot-type analogues in broader polynomial and transcendental families [19] [20]. In this broader context, the present paper remains within the classical quadratic setting, but its emphasis lies at the intersection of exterior conformal uniformization, logarithmic potential theory, and the structural interpretation of escape-rate regularization.

The present article should be read together with [6]. In that work, the principal object was the regularized critical orbit escape-rate function

$\mathcal{G}\left(c\right):=\underset{n\to \infty }{\text{lim}}\frac{1}{2^n}{\text{log}}^{+}\left|f_c^{\circ n}\left(0\right)\right|.$ (20)

The function in (20) was studied as a quantitatively tractable observable of escape. The present article is a geometric companion: the earlier paper studies the effective quantitative behaviour of regularized escape, whereas the present paper identifies the structural analytic object underlying that theory and proves a bridge theorem between the two viewpoints.

3. Preliminaries and Normalizations

Definition 3.1 (Quadratic Family). For $c\in ℂ$ , define

$f_c:ℂ\to ℂ,f_c\left(z\right)=z^2+c.$ (21)

Its $n$ -fold iterate is denoted by $f_c^{\circ n}$ .

Definition 3.2 (Filled Julia set, Julia set, and Basin of Infinity [1], Chapter VI and [5] Chapter 8). For fixed $c\in ℂ$ , the filled Julia set of $f_c$ is

$K_c:=\left\{z\in ℂ:\underset{n\ge 0}{\text{sup}}\left|f_c^{\circ n}\left(z\right)\right|<\infty \right\}.$ (22)

the Julia set is

$J_c:=\partial K_c.$ (23)

and the basin of infinity is

$A_c\left(\infty \right):=ℂ\backslash K_c.$ (24)

Definition 3.3 (Mandelbrot Set, [2] [3] and [5], Chapter 14). The Mandelbrot set is

$ℳ:=\left\{c\in ℂ:\underset{n\ge 0}{\text{sup}}\left|f_c^{\circ n}\left(0\right)\right|<\infty \right\}.$ (25)

Equivalently, $c\in ℳ$ if and only if the critical orbit of $f_c$ is bounded.

Definition 3.4 (Positive Logarithm). For $r\ge 0$ , define

${\log }^{+}r:=\log \max \left\{r,1\right\}.$ (26)

For $r>0$ , the definition is equivalently written as

${\log }^{+}r=\max \left\{\log r,0\right\}.$ (27)

Definition 3.5 (Dynamical Green Function) ([1] Chapter VI and [5] Chapter 8). For each $c\in ℂ$ , the dynamical Green function of $f_c\left(z\right)=z^2+c$ is

$g_c\left(z\right):=\underset{n\to \infty }{\lim }\frac{1}{2^n}{\log }^{+}\left|f_c^{\circ n}\left(z\right)\right|.$ (28)

Here ${\text{log}}^{+}$ is defined in (26). It is a classical theorem that the limit in (28) exists for every $z\in ℂ$ .

Definition 3.6 (Finite Critical Orbit Escape Rate Approximants). For $c\in ℂ$ , let

$z_0=0,z_{n+1}=f_c\left(z_n\right)=z_n^2+c.$ (29)

For $n\ge 0$ , define

$G_n\left(c\right):=\frac{1}{2^n}{\log }^{+}\left|z_n\right|=\frac{1}{2^n}{\log }^{+}\left|f_c^{\circ n}\left(0\right)\right|.$ (30)

The quantities defined in (30) are the finite critical orbit regularizations used later in Theorem 8.1 and in the numerical schemes of Section 8.1. The recurrence (29) is used in the telescoping argument of Theorem 8.1.

By the defining limit (28), evaluated at $z=0$ , and the definition in (30), one has

$g_c\left(0\right)=\underset{n\to \infty }{\text{lim}}{G}_n\left(c\right).$ (31)

Remark 3.7 (Classical Properties of g_c). It is classical that for every $c\in ℂ$ , the function $g_c$ defined in (28) exists on all of $ℂ$ , vanishes precisely on $K_c$ , is harmonic on $A_c\left(\infty \right)$ , and satisfies the functional equation

$g_c\left(f_c\left(z\right)\right)=2g_c\left(z\right).$ (32)

See [1], Chapter VI, or [5], Chapter 8.

Definition 3.8 (Logarithmic Pole at Infinity). Let $\Omega \subset ℂ$ be an unbounded domain whose complement is compact. A real-valued function $u$ on $\Omega$ has a logarithmic pole at infinity with the normalization used in this paper if

$u\left(c\right)-\text{log}\left|c\right|\to 0\left(c\to \infty \right).$ (33)

Equivalently, $u\left(c\right)=\text{log}\left|c\right|+o\left(1\right)$ as $c\to \infty$ . This is the normalization used for the Green function of $ℂ\backslash ℳ$ .

Definition 3.9 (Parameter Green Function, Dynamical Normalization). For $c\in ℂ\backslash ℳ$ , define initially

$G_{ℳ}\left(c\right):=g_c\left(c\right).$ (34)

The dynamical normalization (34) will be identified in Proposition 5.3 with the exterior conformal formula introduced in Definition 5.1.

Remark 3.10 (Normalization Bridge). The normalization in Definition 3.9 is naturally compatible with the parameter-plane Böttcher map introduced in Notation 3.11. It differs by a factor 2 from the critical-point normalization $\mathcal{G}\left(c\right):=g_c\left(0\right)$ used in [6]. Since $f_c\left(0\right)=c$ , the functional Equation (32) gives

$G_{ℳ}\left(c\right)=g_c\left(c\right)=g_c\left(f_c\left(0\right)\right)=2g_c\left(0\right)=2\mathcal{G}\left(c\right),c\in ℂ\backslash ℳ.$ (35)

The identity (35) shows that the two normalizations encode the same escape geometry, but they are adapted to different viewpoints: $\mathcal{G}$ is suited to critical orbit regularization, whereas $G_{ℳ}$ is suited to parameter-plane exterior uniformization.

Notation 3.11 (Parameter Böttcher Map). Throughout the paper, ${\Phi }_{ℳ}$ denotes the conformal map ${\Phi }_{ℳ}:ℂ\backslash ℳ\to ℂ\backslash \overline{\mathbb{D}}$ normalized by

$\frac{{\Phi }_{ℳ}\left(c\right)}{c}\to 1\left(c\to \infty \right).$ (36)

This map is recalled more formally in Theorem 4.8.

Definition 3.12 (Hausdorff Distance). Let $A,B\subset ℂ$ be nonempty compact sets. Their Hausdorff distance is

$d_{\text{H}}\left(A,B\right):=\text{max}\left\{\underset{a\in A}{\text{sup}}\underset{b\in B}{\text{inf}}\left|a-b\right|,\underset{b\in B}{\text{sup}}\underset{a\in A}{\text{inf}}\left|a-b\right|\right\}.$ (37)

The metric (37) is used in Theorem 9.1 to measure stability of equipotentials.

Notation 3.13 (Exterior Circles and Compact Annuli). For $t>0$ , write

$C_t:=\left\{w\in ℂ:\left|w\right|={\text{e}}^t\right\}.$ (38)

For $0<a<b$ , write

$A_{a,b}:=\left\{w\in ℂ:{\text{e}}^a\le \left|w\right|\le {\text{e}}^b\right\}.$ (39)

Remark 3.14 (Compact Subsets of the Exterior). If $K\subset ℂ\backslash ℳ$ is a nonempty compact set, then $K$ lies at a positive Euclidean distance from $ℳ$ . Equivalently,

$\text{dist}\left(K,ℳ\right):=\text{inf}\left\{\left|c-\zeta \right|:c\in K,\zeta \in ℳ\right\}>0.$ (40)

This positive separation does not by itself provide a common escape time for all parameters in $K$ . In the proof of Corollary 8.4, continuity of the critical orbit polynomials from Notation 4.4, together with a finite-subcover argument, is used to construct the common entry index $n_K$ defined in (129).

4. Classical Exterior Uniformization and the Parameter Böttcher Map

We recall the classical Böttcher coordinate near infinity and the corresponding parameter-space uniformization. These results provide the conformal input used in Definition 5.1 and Proposition 5.3.

Theorem 4.1 (Böttcher Coordinate near Infinity) Fix $c\in ℂ$ . There exist a neighbourhood $U_c$ of infinity, a number $R_c>1$ , and a unique conformal map ${\varphi }_c:U_c\to \left\{w\in ℂ:\left|w\right|>R_c\right\}$ such that

${\varphi }_c\left(f_c\left(z\right)\right)={\left({\varphi }_c\left(z\right)\right)}^2.$ (41)

whenever $z,f_c\left(z\right)\in U_c$ , and $\frac{{\varphi }_c\left(z\right)}{z}\to 1$ as $z\to \infty$ . Moreover,

$g_c\left(z\right)=\text{log}\left|{\varphi }_c\left(z\right)\right|\left(z\in U_c\right).$ (42)

Proof. This is the classical Böttcher theorem applied to the super-attracting fixed point at infinity of $f_c\left(z\right)=z^2+c$ ; see [8], Chapter 3, [1], Chapter VI, and [5], Chapter 8. For $z$ sufficiently large,

$f_c\left(z\right)=z^2\left(1+\frac{c}{z^2}\right).$ (43)

The factorization (43) shows that the correction term $c/z^2$ is small near infinity. The standard Böttcher iteration construction produces a holomorphic function ${\varphi }_c$ , defined in an exterior neighbourhood of infinity, satisfying (41) and ${\varphi }_c\left(z\right)=z+O\left(1\right)$ as ($z\to \infty$ ). The stronger normalization $\frac{{\varphi }_c\left(z\right)}{z}\to 1$ determines ${\varphi }_c$ uniquely.

For completeness, the construction can be written locally in the form

${\varphi }_c\left(z\right)=z\exp \left({\displaystyle \sum }_{j=0}^{\infty }\frac{1}{2^{j+1}}\text{Log}\left(1+\frac{c}{{\left(f_c^{\circ j}\left(z\right)\right)}^2}\right)\right).$ (44)

In the representation (44), the neighbourhood of infinity is chosen so that

$\left|\frac{c}{{\left(f_c^{\circ j}\left(z\right)\right)}^2}\right|<1,\forall j\ge 0.$ (45)

Here, $\text{Log}$ denotes the branch satisfying $\text{Log}1=0$ . Condition (45) ensures that the logarithmic terms are evaluated in a common neighbourhood of 1. The series in (44) converges locally uniformly near infinity and therefore defines a holomorphic function there. Finally, by the defining limit (28), the dynamical Green function is obtained from the normalized asymptotic growth of $f_c^{\circ n}\left(z\right)$ . For $z\in U_c$ , the orbit tends to infinity, and the Böttcher functional equation gives

${\varphi }_c\left(f_c^{\circ n}\left(z\right)\right)={\left({\varphi }_c\left(z\right)\right)}^{2^n}.$ (46)

Taking absolute values and logarithms in (46) gives

$\frac{1}{2^n}\text{log}\left|{\varphi }_c\left(f_c^{\circ n}\left(z\right)\right)\right|=\text{log}\left|{\varphi }_c\left(z\right)\right|.$ (47)

Since $\frac{{\varphi }_c\left(w\right)}{w}\to 1$ as ($w\to \infty$ ), the difference between $\text{log}\left|{\varphi }_c\left(f_c^{\circ n}\left(z\right)\right)\right|$ and $\text{log}\left|f_c^{\circ n}\left(z\right)\right|$ tends to zero. Combining the defining limit (28) with (47) therefore proves the identity (42). □

The next result explains why the critical value $c=f_c\left(0\right)$ is the correct parameter-plane point at which to evaluate the dynamical Böttcher coordinate.

Proposition 4.2. If $c\in ℂ\backslash ℳ$ , then both 0 and $c=f_c\left(0\right)$ belong to the basin of infinity $A_c\left(\infty \right)$ from Definition 3.2. Moreover, the critical orbit eventually enters every sufficiently small neighbourhood of infinity. In particular, there exists an integer $N\ge 1$ such that $f_c^{\circ N}\left(0\right)\in U_c$ , where $U_c$ is the exterior neighbourhood on which the local Böttcher coordinate in Theorem 4.1 is defined.

Proof. Assume that $c\notin ℳ$ . By Definition 3.3, the critical orbit $0,f_c\left(0\right)=c,f_c^{\circ 2}\left(0\right),\cdots$ is unbounded.

For a polynomial, an unbounded forward orbit necessarily tends to infinity. Indeed, the reverse triangle inequality gives

$\left|f_c\left(z\right)\right|=\left|z^2+c\right|\ge {\left|z\right|}^2-\left|c\right|.$ (48)

The estimate (48) shows that, once an orbit enters a sufficiently large exterior region, its modulus continues to increase. Consequently, $f_c^{\circ n}\left(0\right)\to \infty$ as $n\to \infty$ . It follows that the critical point 0 does not belong to the filled Julia set $K_c$ . Hence

$0\in A_c\left(\infty \right)=ℂ\backslash K_c.$ (49)

Since $c=f_c\left(0\right)$ , the forward orbit of $c$ is the tail of the critical orbit:

$c,f_c\left(c\right)=f_c^{\circ 2}\left(0\right),f_c^{\circ 2}\left(c\right)=f_c^{\circ 3}\left(0\right),\cdots$ (50)

Therefore,

$f_c^{\circ n}\left(c\right)=f_c^{\circ \left(n+1\right)}\left(0\right)\to \infty .$ (51)

The limit (51) proves that the critical value $c=f_c\left(0\right)$ also belongs to $A_c\left(\infty \right)$ . Finally, because $f_c^{\circ n}\left(0\right)\to \infty$ , the critical orbit eventually enters the exterior neighbourhood $U_c$ supplied by Theorem 4.1. Hence there exists $N\ge 1$ such that $f_c^{\circ N}\left(0\right)\in U_c$ . □

Remark 4.3. Proposition 4.2 shows that the critical value $c=f_c\left(0\right)$ belongs to the basin of infinity and that its forward orbit eventually enters the exterior neighbourhood $U_c$ . It does not, however, imply that the critical value $c$ itself lies in the particular neighbourhood $U_c$ on which the local coordinate of Theorem 4.1 is initially defined. Moreover, Theorem 4.1 is a fixed-parameter statement and does not by itself establish holomorphic dependence of the Böttcher coordinate on $c$ . These two issues are addressed by the family construction near infinity and by the classical Douady–Hubbard parameter uniformization stated below.

Notation 4.4 (Critical orbit Polynomials). For $c\in ℂ$ , define

$z_0\left(c\right):=0,z_{n+1}\left(c\right):=z_n{\left(c\right)}^2+c,n\ge 0.$ (52)

Equivalently, the recurrence (52) gives

$z_n\left(c\right)=f_c^{\circ n}\left(0\right).$ (53)

For every $n\ge 1$ , the function $c\mapsto z_n\left(c\right)$ is a polynomial in $c$ . In particular,

$z_1\left(c\right)=c,z_2\left(c\right)=c^2+c,z_3\left(c\right)={\left(c^2+c\right)}^2+c.$ (54)

If $c\in ℂ\backslash ℳ$ , then $z_n\left(c\right)\to \infty$ as $n\to \infty$ by Proposition 4.2.

Definition 4.5 (Local Parameter Böttcher Branch). Let $V\subset ℂ\backslash ℳ$ be a simply connected parameter domain. Suppose that $N\ge 1$ and $R>1$ are such that $\left|z_N\left(c\right)\right|>R$ for $c\in V$ , where $z_N\left(c\right)=f_c^{\circ N}\left(0\right)$ .

Let $\varphi :V\times \{z\in ℂ:\left|z\right|>R\}\to ℂ$ be the jointly holomorphic Böttcher coordinate supplied by Proposition 4.7. A local parameter Böttcher branch on $V$ is a holomorphic function ${\Phi }_V:V\to ℂ\backslash \overline{\mathbb{D}}$ , satisfying

${\Phi }_V{\left(c\right)}^{2^{N-1}}=\varphi \left(c,z_N\left(c\right)\right),c\in V.$ (55)

Equation (55) is understood with the root branch selected compatibly with the normalization (36).

Notation 4.6 (Normalized Critical orbit Root Approximants). On a simply connected parameter domain $V\subset ℂ\backslash ℳ$ where compatible holomorphic roots have been selected, write

${\Phi }_n\left(c\right):=z_n{\left(c\right)}^{1/2^{n-1}},n\ge 1.$ (56)

The root expressions in (56) are understood locally through compatible holomorphic branches. This notation is local until the global parameter Böttcher map has been established by Theorem 4.8.

The following standard family version of the local Böttcher construction is included with proof in the notation required below; compare [1], Chapter VI and [5], Chapter 8.

Proposition 4.7 (Joint Holomorphic Dependence near Infinity). Let $V\subset ℂ$ be a relatively compact parameter domain. Then there exists $R>1$ and a jointly holomorphic function $\varphi :V\times \left\{z\in ℂ:\left|z\right|>R\right\}\to ℂ$ such that, for every $c\in V$ , the map $z\mapsto {\varphi }_c\left(z\right):=\varphi \left(c,z\right)$ is the normalized Böttcher coordinate of $f_c$ on $\left\{\left|z\right|>R\right\}$ . In particular, $\varphi \left(c,f_c\left(z\right)\right)=\varphi {\left(c,z\right)}^2$ whenever $\left|z\right|>R$ , and $\frac{\varphi \left(c,z\right)}{z}\to 1$ as $z\to \infty$ , locally uniformly with respect to $c\in V$ .

Proof. Since $V$ is relatively compact, its closure $\overline{V}$ is compact. Hence there exists $B>0$ such that $\left|c\right|\le B$ for every $c\in \overline{V}$ , and therefore for every $c\in V$ . Choose $R>1$ sufficiently large that $R^2-B>R$ and $\frac{B}{R^2}<\frac{1}{2}$ . Then, for $c\in V$ and $\left|z\right|>R$ ,

$\left|f_c\left(z\right)\right|=\left|z^2+c\right|\ge {\left|z\right|}^2-\left|c\right|>R.$ (57)

Estimate (57) shows that the common exterior region $\left\{z\in ℂ:\left|z\right|>R\right\}$ is forward invariant under $f_c$ for every $c\in V$ . Consequently, for $c\in V$ and $\left|z\right|>R$ , define $z_j\left(c,z\right):=f_c^{\circ j}\left(z\right)$ . By (57),

$\left|z_j\left(c,z\right)\right|>R,j\ge 0.$ (58)

For $c\in V$ and $\left|z\right|>R$ , define

$H\left(c,z\right):={\displaystyle \sum }_{j=0}^{\infty }\frac{1}{2^{j+1}}\text{Log}\left(1+\frac{c}{z_j{\left(c,z\right)}^2}\right),$ (59)

where $\text{Log}$ denotes the branch of the logarithm defined near 1 and normalized by $\text{Log}1=0$ .

Since $\left|c\right|\le B$ , the orbit estimate (58) and the choice $B/R^2<1/2$ give

$\left|\frac{c}{z_j{\left(c,z\right)}^2}\right|\le \frac{B}{R^2}<\frac{1}{2},j\ge 0.$ (60)

Condition (60) ensures that every logarithmic term in (59) is evaluated in a common neighbourhood of 1. Moreover, the forward-invariance estimate (57) implies rapid growth of the iterates in the common exterior region. Consequently, the series in (59) converges locally uniformly on $V\times \left\{z\in ℂ:\left|z\right|>R\right\}$ . Each summand is jointly holomorphic in $\left(c,z\right)$ , and therefore $H\left(c,z\right)$ is jointly holomorphic.

Define

$\varphi \left(c,z\right):=z\exp \left(H\left(c,z\right)\right).$ (61)

Since $H$ is jointly holomorphic, the definition (61) shows that $\varphi$ is jointly holomorphic.

Separating the first term of the series (59) and using the recurrence $z_{j+1}\left(c,z\right)=z_j\left(c,f_c\left(z\right)\right)$ gives

$\varphi \left(c,f_c\left(z\right)\right)=\varphi {\left(c,z\right)}^2.$ (62)

By (61), $\frac{\varphi \left(c,z\right)}{z}=\text{exp}\left(H\left(c,z\right)\right)$ . Since the series (59) tends locally uniformly to zero as $z\to \infty$ , one obtains

$\frac{\varphi \left(c,z\right)}{z}\to 1\left(z\to \infty \right),$ (63)

locally uniformly with respect to $c\in V$ .

For each fixed $c\in V$ , the functional Equation (62) and the normalization (63), together with uniqueness in Theorem 4.1, show that $\varphi \left(c,\cdot \right)$ is the normalized local Böttcher coordinate of $f_c$ . □

Theorem 4.8 (Douady--Hubbard Parameter Uniformization; classical, see [2] [3], [4] Chapter 13, and [5] Chapter 14). There exists a unique conformal isomorphism ${\Phi }_{ℳ}:ℂ\backslash ℳ\to ℂ\backslash \overline{\mathbb{D}}$ satisfying the normalization $\frac{{\Phi }_{ℳ}\left(c\right)}{c}\to 1$ as $c\to \infty$ . Let $c_0\in ℂ\backslash ℳ$ . Then there exist a simply connected neighbourhood $V\subset ℂ\backslash ℳ$ of $c_0$ , an integer $N\ge 1$ , and a number $R>1$ such that $\left|z_N\left(c\right)\right|>R$ , $c\in V$ where $z_N\left(c\right)=f_c^{\circ N}\left(0\right)$ is the critical orbit polynomial defined in Notation 4.4. Moreover, if $\varphi :V\times \left\{z\in ℂ:\left|z\right|>R\right\}\to ℂ$ is the jointly holomorphic exterior Böttcher coordinate supplied by Proposition 4.7, then

${\Phi }_{ℳ}{\left(c\right)}^{2^{N-1}}=\varphi \left(c,z_N\left(c\right)\right),c\in V.$ (64)

The relation (64) identifies the restriction ${{\Phi }_{ℳ}|}_V$ with the local parameter Böttcher branch of Definition 4.5 that is compatible with the normalization at infinity. In particular, ${\Phi }_{ℳ}$ is holomorphic and nonvanishing on $ℂ\backslash ℳ$ , and

$\left|{\Phi }_{ℳ}\left(c\right)\right|>1,c\in ℂ\backslash ℳ.$ (65)

Proof. The existence of the conformal isomorphism ${\Phi }_{ℳ}:ℂ\backslash ℳ\to ℂ\backslash \overline{\mathbb{D}}$ , with the normalization (36), is the classical Douady–Hubbard external uniformization theorem for the quadratic connectedness locus.

We verify the stated local representation in the notation used in this paper. Fix $c_0\in ℂ\backslash ℳ$ . By Proposition 4.2, and the critical orbit representation (53),

$z_n\left(c_0\right)=f_{c_0}^{\circ n}\left(0\right)\to \infty \left(n\to \infty \right).$ (66)

Choose a relatively compact parameter neighbourhood $W\subset ℂ\backslash ℳ$ of $c_0$ . By Proposition 4.7, there exist $R>1$ and a jointly holomorphic function $\varphi :W\times \left\{z\in ℂ:\left|z\right|>R\right\}\to ℂ$ such that, for each $c\in W$ , the function ${\varphi }_c\left(z\right):=\varphi \left(c,z\right)$ is the normalized dynamical Böttcher coordinate of $f_c$ on the common exterior region $\left\{\left|z\right|>R\right\}$ . By Proposition 4.7, the map $\varphi$ satisfies the functional Equation (62) and the normalization (63). Since $z_n\left(c_0\right)\to \infty$ , there exists $N\ge 1$ such that $\left|z_N\left(c_0\right)\right|>R$ . By Notation 4.4, $z_N\left(c\right)$ is a polynomial in $c$ , and hence is holomorphic and continuous. After shrinking $W$ to a sufficiently small simply connected neighbourhood $V$ of $c_0$ , we therefore have

$\left|z_N\left(c\right)\right|>R,c\in V.$ (67)

Define

$F_N\left(c\right):=\varphi \left(c,z_N\left(c\right)\right).$ (68)

By (67), the composition in (68) is holomorphic on $V$ . Moreover, because each ${\varphi }_c$ is a Böttcher coordinate with values in $ℂ\backslash \overline{\mathbb{D}}$ , the function $F_N$ is nonvanishing on $V$ .

Because the function $F_N$ defined in (68) is nonvanishing on the simply connected domain $V$ , it admits a holomorphic logarithm and, consequently, a holomorphic $2^{N-1}$ -st root. By Definition 4.5, the root branch compatible with the normalization at infinity defines ${\Phi }_V$ and satisfies (55).

The definition is independent of the chosen entry time. Indeed, if $m\ge N$ , then

$z_m\left(c\right)=f_c^{\circ \left(m-N\right)}\left(z_N\left(c\right)\right).$ (69)

Repeated application of the functional Equation (62) along the orbit identity (69) gives

$\varphi \left(c,z_m\left(c\right)\right)=\varphi {\left(c,z_N\left(c\right)\right)}^{2^{m-N}}.$ (70)

Therefore, for the compatible local root branches,

$\varphi {\left(c,z_m\left(c\right)\right)}^{1/2^{m-1}}=\varphi {\left(c,z_N\left(c\right)\right)}^{1/2^{N-1}}.$ (71)

Identity (71) proves that the normalized pullback is unchanged when a later entry time is used.

The classical Douady–Hubbard theorem guarantees that the normalized local branches obtained in this manner are compatible on overlaps and patch together to form a single-valued holomorphic map on $ℂ\backslash ℳ$ . This global map is precisely ${\Phi }_{ℳ}$ . Consequently, the globally defined parameter Böttcher map satisfies the local representation (64) on $V$ .

The same classical theorem establishes that ${\Phi }_{ℳ}$ is one-to-one and maps $ℂ\backslash ℳ$ onto $ℂ\backslash \overline{\mathbb{D}}$ . Therefore ${\Phi }_{ℳ}$ is a conformal isomorphism. Since ${\Phi }_{ℳ}$ maps $ℂ\backslash ℳ$ onto $ℂ\backslash \overline{\mathbb{D}}$ , the modulus estimate (65) follows.

It remains to prove uniqueness under the stated normalization. Suppose that $\Psi :ℂ\backslash ℳ\to ℂ\backslash \overline{\mathbb{D}}$ is another conformal isomorphism satisfying the normalization $\Psi \left(c\right)/c\to 1$ as $c\to \infty$ , analogous to (36). Then $A:=\Psi \circ {\Phi }_{ℳ}^{-1}$ is an automorphism of $ℂ\backslash \overline{\mathbb{D}}$ that fixes infinity. Every such automorphism is a rotation, $A\left(w\right)={\text{e}}^{i\theta }w$ . The normalization at infinity gives ${\text{e}}^{i\theta }=1$ . Hence $A$ is the identity map, and therefore $\Psi ={\Phi }_{ℳ}$ . □

Corollary 4.9 (Critical Orbit Representation). Let $V\subset ℂ\backslash ℳ$ be a simply connected parameter neighbourhood on which the compatible holomorphic root branches are chosen according to Definition 4.5. Then

${\Phi }_{ℳ}\left(c\right)=\underset{n\to \infty }{\lim }{z}_n{\left(c\right)}^{1/2^{n-1}},$ (72)

locally uniformly for $c\in V$ . Equivalently,

${\Phi }_{ℳ}\left(c\right)=\underset{n\to \infty }{\lim }{\left(f_c^{\circ n}\left(0\right)\right)}^{1/2^{n-1}}.$ (73)

where the roots denote the local holomorphic branches compatible with the normalization (36).

Proof. Let $N$ and $R$ be as in Theorem 4.8. For $n\ge N$ , the orbit point $z_n\left(c\right)$ lies in the common exterior Böttcher region. Since

$\varphi \left(c,z\right)=z\left(1+o\left(1\right)\right)\left(z\to \infty \right),$ (74)

locally uniformly with respect to $c$ , applying (74) along the escaping critical orbit gives

$\frac{\varphi \left(c,z_n\left(c\right)\right)}{z_n\left(c\right)}\to 1\left(n\to \infty \right),$ (75)

locally uniformly on $V$ . By the functional equation and Theorem 4.8,

$\varphi \left(c,z_n\left(c\right)\right)={\Phi }_{ℳ}{\left(c\right)}^{2^{n-1}}.$ (76)

Consequently,

$z_n{\left(c\right)}^{1/2^{n-1}}={\Phi }_{ℳ}\left(c\right){\left(\frac{z_n\left(c\right)}{\varphi \left(c,z_n\left(c\right)\right)}\right)}^{1/2^{n-1}}.$ (77)

By (75), the second factor in (77) converges locally uniformly to $1$ . Hence the convergence asserted in (72) follows. □

Remark 4.10. The root-limit formula in Corollary 4.9 is not used as a branch-free global definition of ${\Phi }_{ℳ}$ . For arbitrary root choices, the expressions $z_n{\left(c\right)}^{1/2^{n-1}}$ are multivalued and do not by themselves establish either single-valuedness or holomorphic dependence on $c$ . The global single-valued holomorphic map ${\Phi }_{ℳ}$ is supplied by the classical Douady–Hubbard uniformization theorem. The corollary expresses this already-established map locally through the uniquely compatible holomorphic root branches determined by its normalization at infinity.

Remark 4.11. The notation ${\Phi }_{ℳ}\left(c\right)={\varphi }_c\left(c\right)$ is commonly used as shorthand for the classical identification between the parameter Böttcher map and the normalized dynamical exterior coordinate evaluated along the escaping critical orbit. It must not be interpreted as a direct evaluation of the local map supplied by Theorem 4.1, because the critical value $c$ need not belong to the particular neighbourhood $U_c$ on which that local coordinate is initially defined. More precisely, if $N\ge 1$ is chosen so that $z_N\left(c\right)=f_c^{\circ N}\left(0\right)$ lies in the exterior Böttcher region, then

${\Phi }_{ℳ}{\left(c\right)}^{2^{N-1}}={\varphi }_c\left(z_N\left(c\right)\right).$ (78)

with the root selected by the normalization at infinity. Consequently,

$\log \left|{\Phi }_{ℳ}\left(c\right)\right|=\frac{1}{2^{N-1}}\log \left|{\varphi }_c\left(z_N\left(c\right)\right)\right|.$ (79)

Taking absolute values and logarithms in (78) gives (79). By the local Green-function identity (42), evaluated at $z_N\left(c\right)$ ,

$g_c\left(z_N\left(c\right)\right)=\text{log}\left|{\varphi }_c\left(z_N\left(c\right)\right)\right|.$ (80)

Combining (79), (80), and the dynamical functional Equation (32) gives

$\log \left|{\Phi }_{ℳ}\left(c\right)\right|=g_c\left(c\right)=2g_c\left(0\right).$ (81)

The geometric content of Theorem 4.8 is summarized in Figure 1. The exterior coordinate sends equipotentials to concentric circles and parameter external rays to radial half-lines.

Figure 1. Schematic representation of the classical Douady–Hubbard exterior uniformization. By Theorem 4.8, the parameter Böttcher map ${\Phi }_{ℳ}$ conformally identifies $ℂ\backslash ℳ$ with $ℂ\backslash \overline{\mathbb{D}}$ . Under this map, the equipotential $G_{ℳ}\left(c\right)=t$ , defined using Definition 5.1, corresponds to the circle $\left|w\right|={\text{e}}^t$ , as stated in Proposition 7.2. The parameter external ray of angle $\theta$ , introduced in Definition 7.1, corresponds to the radial half-line $w=r{\text{e}}^{2\pi \text{i}\theta }$ , $r>1$ . The parameter-plane ray is schematic and is included to illustrate the conformal correspondence rather than to represent a numerically computed inverse Böttcher ray.

5. Explicit Construction of the Green Function

We now construct the Green function of $ℂ\backslash ℳ$ explicitly from the parameter Böttcher map of Notation 3.11 and Theorem 4.8. The results of this section are classical consequences of the Douady-Hubbard parameter uniformization and standard logarithmic potential theory; see [1], Chapter VI, [2] [3], [5], Chapter 14, and [9]. They are included with proofs to fix the precise normalization used in the quantitative analysis.

Definition 5.1 (Exterior Green function). Let ${\Phi }_{ℳ}:ℂ\backslash ℳ\to ℂ\backslash \overline{\mathbb{D}}$ be the normalized parameter Böttcher map from Theorem 4.8. Define

$G_{ℳ}\left(c\right):=\log \left|{\Phi }_{ℳ}\left(c\right)\right|,c\in ℂ\backslash ℳ.$ (82)

Equation (82) is the exact conformal definition used throughout the remainder of the paper.

Remark 5.2. Definition 5.1 is the classical conformal representation of the exterior Green function:

$G_{ℳ}=\log |\cdot |\circ {\Phi }_{ℳ}.$ (83)

The composition formula (83) shows that $G_{ℳ}$ is defined exactly on $ℂ\backslash ℳ$ , rather than only through its asymptotic behaviour at infinity.

We next identify this parameter-plane function with the dynamical Green function evaluated at the critical value.

Proposition 5.3. For every $c\in ℂ\backslash ℳ$ , $G_{ℳ}\left(c\right)=g_c\left(c\right).$

Proof. Fix $c\in ℂ\backslash ℳ$ , and write $z_n\left(c\right)=f_c^{\circ n}\left(0\right)$ . Choose $N\ge 1$ so that $z_N\left(c\right)$ belongs to an exterior neighbourhood on which the dynamical Böttcher coordinate ${\varphi }_c$ is defined. By Theorem 4.8, the local representation (64),

${\Phi }_{ℳ}{\left(c\right)}^{2^{N-1}}={\varphi }_c\left(z_N\left(c\right)\right).$ (84)

Taking absolute values and logarithms in (84) gives

$\log \left|{\Phi }_{ℳ}\left(c\right)\right|=\frac{1}{2^{N-1}}\log \left|{\varphi }_c\left(z_N\left(c\right)\right)\right|.$ (85)

By Theorem 4.1, and the local identity (42),

$\text{log}\left|{\varphi }_c\left(z_N\left(c\right)\right)\right|=g_c\left(z_N\left(c\right)\right).$ (86)

Since the critical value is $c=f_c\left(0\right)$ , one has

$z_N\left(c\right)=f_c^{\circ \left(N-1\right)}\left(c\right).$ (87)

Repeated application of the functional Equation (32) along (87) gives

$g_c\left(z_N\left(c\right)\right)=2^{N-1}{g}_c\left(c\right).$ (88)

Combining (85), (86), and (88) yields

$\log \left|{\Phi }_{ℳ}\left(c\right)\right|=g_c\left(c\right).$ (89)

Using Definition 5.1 and (82), we obtain $G_{ℳ}\left(c\right)=g_c\left(c\right)$ . □

Corollary 5.4. For every $c\in ℂ\backslash ℳ$ , $G_{ℳ}\left(c\right)=2g_c\left(0\right)$ . Consequently,

$G_{ℳ}\left(c\right)=2\underset{n\to \infty }{\text{lim}}\frac{1}{2^n}{\text{log}}^{+}\left|f_c^{\circ n}\left(0\right)\right|.$ (90)

Proof. Since $c=f_c\left(0\right)$ , the functional Equation (32) from Remark 3.7 gives

$g_c\left(c\right)=g_c\left(f_c\left(0\right)\right)=2g_c\left(0\right).$ (91)

Using Proposition 5.3, we conclude that $G_{ℳ}\left(c\right)=2g_c\left(0\right)$ . The limit formula (90) then follows from Definition 3.5 and (28), evaluated at $z=0$ . □

Remark 5.5. Corollary 5.4 is the precise normalization bridge between the present paper and [6]. The earlier quantitative paper studied the critical-point regularization $g_c\left(0\right)$ , whereas the present paper is organized around the parameter-plane Green function $G_{ℳ}\left(c\right)=g_c\left(c\right)$ . The factor 2 reflects the single-step passage from the critical point 0 to the critical value $c=f_c\left(0\right)$ .

Proposition 5.6. The function $G_{ℳ}$ is harmonic on $ℂ\backslash ℳ$ .

Proof. By Theorem 4.8, the map ${\Phi }_{ℳ}$ is holomorphic and maps $ℂ\backslash ℳ$ into $ℂ\backslash \overline{\mathbb{D}}$ . In particular, ${\Phi }_{ℳ}$ is nowhere zero on $ℂ\backslash ℳ$ . Hence, near every point of $ℂ\backslash ℳ$ , one may choose a holomorphic branch of $\text{Log}{\Phi }_{ℳ}$ . On such a neighbourhood,

$\log \left|{\Phi }_{ℳ}\right|=\mathrm{Re}\left(\text{Log}{\Phi }_{ℳ}\right).$ (92)

Identity (92) shows that $\log \left|{\Phi }_{ℳ}\right|$ is harmonic locally. Since harmonicity is a local property and $G_{ℳ}=\log \left|{\Phi }_{ℳ}\right|$ by Definition 5.1 and (82), $G_{ℳ}$ is harmonic on $ℂ\backslash ℳ$ . □

Proposition 5.7. For every $c\in ℂ\backslash ℳ$ , one has $G_{ℳ}\left(c\right)>0.$

Proof. If $c\in ℂ\backslash ℳ$ , then Theorem 4.8 gives ${\Phi }_{ℳ}\left(c\right)\in ℂ\backslash \overline{\mathbb{D}}$ . The modulus estimate (65) gives $\left|{\Phi }_{ℳ}\left(c\right)\right|>1$ . Taking logarithms and using Definition 5.1 together with (82) gives $G_{ℳ}\left(c\right)>0$ . □

Theorem 5.8 (Logarithmic growth at infinity). As $c\to \infty$ ,

$G_{ℳ}\left(c\right)-\text{log}\left|c\right|\to 0.$ (93)

Proof. By Theorem 4.8 and the normalization (36), $\frac{{\Phi }_{ℳ}\left(c\right)}{c}\to 1$ as $c\to \infty$ . Taking moduli gives

$\left|\frac{{\Phi }_{ℳ}\left(c\right)}{c}\right|\to 1\left(c\to \infty \right).$ (94)

Since the real logarithm is continuous at 1, it follows that

$\log \left|\frac{{\Phi }_{ℳ}\left(c\right)}{c}\right|=\log \left|{\Phi }_{ℳ}\left(c\right)\right|-\log \left|c\right|\to 0.$ (95)

Using (82) in (95) gives (93). □

Corollary 5.9. The function $G_{ℳ}$ has a logarithmic pole at infinity in the sense of Definition 3.8.

Proof. By Theorem 5.8 and (93), $G_{ℳ}\left(c\right)=\text{log}\left|c\right|+o\left(1\right)$ as $c\to \infty$ , which is exactly the normalization in Definition 3.8 and (33). □

Proposition 5.10. The function $G_{ℳ}$ extends continuously to $ℂ$ by setting $G_{ℳ}\left(c\right)=0$ for all $c\in ℳ$ . In particular, $G_{ℳ}=0$ on $\partial ℳ$ .

Proof. By Corollary 5.4, $G_{ℳ}\left(c\right)=2g_c\left(0\right)$ for all $c\in ℂ\backslash ℳ$ . The escape-rate function $\left(c,z\right)\mapsto g_c\left(z\right)$ is continuous for the quadratic family; see, for example, [1], Chapter VI, or [5], Chapter 8. In particular, the function $c\mapsto g_c\left(0\right)$ is continuous on $ℂ$ . Moreover,

$g_c\left(0\right)=0\iff 0\in K_c\iff c\in ℳ.$ (96)

Let $c_j\in ℂ\backslash ℳ$ and suppose that $c_j\to c_{*}\in \partial ℳ$ . By continuity,

$G_{ℳ}\left(c_j\right)=2g_{c_j}\left(0\right)\to 2g_{c_{*}}\left(0\right)=0.$ (97)

The final equality in (97) follows from (96), since $c_{*}\in \partial ℳ\subset ℳ$ . Define

$G_{ℳ}\left(c\right):=0,c\in ℳ.$ (98)

The convergence (97) shows that the extension (98) is continuous on all of $ℂ$ . In particular, $G_{ℳ}=0$ on $\partial ℳ$ . □

Theorem 5.11 (Structural Characterization of $G_{ℳ}$ ). The function $G_{ℳ}$ constructed in Definition 5.1 has the following properties:

(i) $G_{ℳ}$ is harmonic on $ℂ\backslash ℳ$ ;

(ii) $G_{ℳ}\left(c\right)>0$ for every $c\in ℂ\backslash ℳ$ ;

(iii) $G_{ℳ}$ extends continuously to $ℂ$ and vanishes on $ℳ$ ;

(iv) $G_{ℳ}\left(c\right)-\text{log}\left|c\right|\to 0$ as $c\to \infty$ .

Proof. Part (i) is Proposition 5.6. Part (ii) is Proposition 5.7. Part (iii) is Proposition 5.10. Part (iv) is Theorem 5.8, equivalently the logarithmic-pole normalization of Definition 3.8. □

6. Uniqueness of the Exterior Green Function

We now prove the classical uniqueness characterization of $G_{ℳ}$ as the Green function of $ℂ\backslash ℳ$ with pole at infinity in the normalization of Definition 3.8; compare [9] and [10]. The proof is included for completeness.

Theorem 6.1 (Uniqueness of the Normalized Exterior Green Function) Let $u:ℂ\backslash ℳ\to ℝ$ be a function satisfying:

(a) $u$ is harmonic on $ℂ\backslash ℳ$ ;

(b) $u$ extends continuously to $\partial ℳ$ and $u=0$ on $\partial ℳ$ ;

(c) $u$ has a logarithmic pole at infinity with the normalization of Definition 3.8, namely the condition (33).

Then $u\left(c\right)=G_{ℳ}\left(c\right)$ for every $c\in ℂ\backslash ℳ.$

Proof. Define

$h\left(c\right):=u\left(c\right)-G_{ℳ}\left(c\right),c\in ℂ\backslash ℳ.$ (99)

By the harmonicity of $u$ and Proposition 5.6, the function $h$ defined in (99) is harmonic on $ℂ\backslash ℳ$ .

By the boundary condition on $u$ and Proposition 5.10, the function $h$ extends continuously to $\partial ℳ$ and satisfies $h=0$ on $\partial ℳ$ . Furthermore, by the growth assumption on $u$ , Theorem 5.8, and (93),

$h\left(c\right)=\left(u\left(c\right)-\text{log}\left|c\right|\right)-\left(G_{ℳ}\left(c\right)-\text{log}\left|c\right|\right)\to 0\left(c\to \infty \right).$ (100)

Fix $c_0\in ℂ\backslash ℳ$ . Since $ℳ$ is compact, choose $R_0>0$ such that $ℳ\subset \left\{\left|c\right|<R_0\right\}$ . For $R>R_0$ with $\left|c_0\right|<R$ , set

${\Omega }_R:=\left\{c\in ℂ\backslash ℳ:\left|c\right|<R\right\}.$ (101)

The function $h$ is harmonic on ${\Omega }_R$ , continuous up to the boundary pieces $\partial ℳ$ and $\left\{\left|c\right|=R\right\}$ , and the boundary of ${\Omega }_R$ is contained in

$\partial {\Omega }_R\subset \partial ℳ\cup \left\{c\in ℂ:\left|c\right|=R\right\}.$ (102)

Set

$\epsilon \left(R\right):=\underset{\left|c\right|=R}{\text{sup}}\left|h\left(c\right)\right|.$ (103)

By (100), $\epsilon \left(R\right)\to 0$ as $R\to \infty$ . The maximum principle applied to both $h$ and $-h$ on ${\Omega }_R$ gives

$\left|h\left(c\right)\right|\le \max \left\{\underset{\partial ℳ}{\sup }\left|h\right|,\underset{\left|c\right|=R}{\sup }\left|h\right|\right\}=\epsilon \left(R\right),c\in {\Omega }_R.$ (104)

Applying (104) at $c=c_0$ gives $\left|h\left(c_0\right)\right|\le \epsilon \left(R\right)$ . Letting $R\to \infty$ and using $\epsilon \left(R\right)\to 0$ , which follows from (100), gives $h\left(c_0\right)=0$ . Since $c_0\in ℂ\backslash ℳ$ was arbitrary, it follows that $h\equiv 0$ on $ℂ\backslash ℳ$ . Therefore, $u=G_{ℳ}$ on $ℂ\backslash ℳ$ . □

Corollary 6.2. The function $G_{ℳ}$ is the normalized Green function of $ℂ\backslash ℳ$ with pole at infinity. Moreover, it is unique among functions satisfying the hypotheses of Theorem 6.1.

Proof. By Theorem 5.8 and (93), the function $G_{ℳ}$ has the normalization required by Definition 3.8 and (33). Thus $G_{ℳ}$ satisfies the existence properties required of the normalized Green function on $ℂ\backslash ℳ$ .

If $u:ℂ\backslash ℳ\to ℝ$ is any other function with the same normalization at infinity and satisfying the hypotheses of Theorem 6.1, then Theorem 6.1 gives $u=G_{ℳ}$ on $ℂ\backslash ℳ.$ This proves uniqueness. □

Remark 6.3. The proof of Theorem 6.1 is an unbounded-domain uniqueness argument. The difference of two admissible Green functions is harmonic, vanishes on the inner boundary, tends to zero on the outer boundary at infinity, and is therefore forced to vanish identically by an exhaustion of the domain and the maximum principle.

7. Equipotentials and External Geometry

The conformal representation (82) from Definition 5.1 makes the geometry of the Mandelbrot exterior transparent. The following standard equipotential and external-ray conventions are induced by the parameter Böttcher coordinate; see [2] [3], [4] Chapter 13, and [5] Chapter 14.

Definition 7.1 (Equipotentials and External Rays). For $t>0$ , the equipotential of level $t$ is

$E_t:=\left\{c\in ℂ\backslash ℳ:G_{ℳ}\left(c\right)=t\right\},t>0.$ (105)

For $\theta \in ℝ/\mathbb{Z}$ , the parameter external ray of angle $\theta$ , measured in turns, is

${ℛ}_{\theta }:={\Phi }_{ℳ}^{-1}\left(\left\{r{\text{e}}^{2\pi \text{i}\theta }:r>1\right\}\right),\theta \in ℝ/\mathbb{Z}.$ (106)

Proposition 7.2. For every $t>0$ , $E_t={\Phi }_{ℳ}^{-1}\left(C_t\right)$ , where $C_t$ is the exterior circle defined in (38) and Notation 3.13.

Proof. Let $t>0$ , and let $c\in ℂ\backslash ℳ$ . By Definition 7.1, Definition 5.1, and Notation 3.13, together with (105), (82), and (38), one has

$\begin{array}{l}c\in E_t\iff G_{ℳ}\left(c\right)=t\\ \iff \log \left|{\Phi }_{ℳ}\left(c\right)\right|=t\\ \iff \left|{\Phi }_{ℳ}\left(c\right)\right|=e^t\\ \iff {\Phi }_{ℳ}\left(c\right)\in C_t.\end{array}$ (107)