Description
Douady-Hubbard Theorem (1981-82): The Mandelbrot set M is connected. Proof: Construct φ from complement of M to complement of closed unit disk by φ(c) = lim (f_cⁿ(c))^(1/2ⁿ) (Böttcher coordinate in parameter space). Show φ is analytic (Böttcher-Fatou), proper, open, surjective, injective (argument principle). Then φ is biholomorphism so complement of M is simply connected so M is connected. Orsay Notes document the full theory.
Source: Wikipedia; Douady and Hubbard; Orsay Notes
Dependency Flowchart
graph TD
Bott["Böttcher and External Rays"]
DefPhi["Def: Douady-Hubbard map φ"]
LemAnalytic["Lem: φ analytic on complement of M"]
LemProper["Lem: φ proper"]
LemOpen["Lem: φ open"]
LemSurj["Lem: φ surjective onto complement of disk"]
LemInj["Lem: φ injective Argument principle"]
ThmConn["Thm: M connected φ biholo implies complement simply conn"]
ThmSimp["Thm: M simply connected Carathéodory later"]
Bott --> DefPhi
DefPhi --> LemAnalytic
LemAnalytic --> LemProper
LemAnalytic --> LemOpen
LemAnalytic --> LemSurj
LemAnalytic --> LemInj
LemInj --> ThmConn
LemSurj --> ThmConn
LemOpen --> ThmConn
ThmConn --> ThmSimp
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 Bott axiom
class DefPhi definition
class ThmConn,ThmSimp theorem
class LemAnalytic,LemProper,LemOpen,LemSurj,LemInj lemma
Color Scheme
Red Prerequisite
Blue Definitions
Teal Theorems
Dark Red Lemmas
Process Statistics
- Nodes: 10
- Edges: 14
- Axioms: 1
- Definitions: 1
- Lemmas: 5
- Theorems: 2