TITLE:
The Exterior Green Function of the Mandelbrot Set: Conformal Construction and Quantitative Escape-Rate Approximation
AUTHORS:
Abhirup Moitra
KEYWORDS:
Mandelbrot Set, Green Function, Böttcher Coordinate, Exterior Uniformization, Escape-Rate Approximation, Equipotentials
JOURNAL NAME:
Open Journal of Modelling and Simulation,
Vol.14 No.4,
September
23,
2026
ABSTRACT: For the quadratic family
f
c
(
z
)=
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
Φ
ℳ
:ℂ\ℳ→ℂ\
D
¯
, normalized at infinity, and the representation
G
ℳ
(
c
)=log|
Φ
ℳ
(
c
) |
. The harmonicity and positivity of
G
ℳ
on
ℂ\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
(
c
)=
g
c
(
c
)=2
g
c
(
0
)
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(
2
−n
)
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\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.