Description
Böttcher coordinate: near infinity, f_c(z) = z² + c is conjugate to w maps to w². Conformal map Φ_c on basin of infinity satisfying Φ_c(f_c(z)) = (Φ_c(z))². Equipotentials: level sets |Φ_c(z)| = const. External rays: curves arg(Φ_c(z)) = θ. For parameter space: Φ(c) = lim (f_cⁿ(c))^(1/2ⁿ) defines Böttcher map on complement of M. External rays of M: R_M(θ) = {c : arg Φ(c) = 2πθ}. Key for Douady-Hubbard connectedness proof.
Source: Wikipedia; Böttcher; Douady-Hubbard
Dependency Flowchart
graph TD
Quad["Quadratic Julia and Mandelbrot"]
DefBott["Def: Böttcher coordinate Φ_c"]
DefFuncEq["Def: Functional equation Φ_c(f_c(z)) = (Φ_c(z))²"]
DefEquip["Def: Equipotential |Φ_c(z)| = const"]
DefExtRay["Def: External ray R(θ) arg Φ_c(z) = 2πθ"]
DefPhiParam["Def: Parameter Böttcher Φ(c) lim (f_cⁿ(c))^(1/2ⁿ)"]
ThmBott["Thm: Böttcher-Fatou Φ analytic on basin of infinity"]
LemLand["Lem: Ray landing Rays land on J_c or M"]
Quad --> DefBott
DefBott --> DefFuncEq
DefBott --> DefEquip
DefBott --> DefExtRay
DefBott --> DefPhiParam
DefPhiParam --> ThmBott
DefExtRay --> LemLand
classDef axiom fill:#ff6b6b,color:#fff,stroke:#c0392b
classDef definition fill:#b197fc,color:#fff,stroke:#9775fa
classDef theorem fill:#51cf66,color:#fff,stroke:#40c057
classDef lemma fill:#74c0fc,color:#fff,stroke:#4dabf7
class Quad axiom
class DefBott,DefFuncEq,DefEquip,DefExtRay,DefPhiParam definition
class ThmBott theorem
class LemLand lemma
Color Scheme
Red Prerequisite
Blue Definitions
Teal Theorems
Purple Lemmas
Process Statistics
- Nodes: 9
- Edges: 10
- Axioms: 1
- Definitions: 5
- Lemmas: 1
- Theorems: 1