Description
Polynomial-like map: f: U → V with U ⋐ V, f proper, degree d. Douady-Hubbard straightening theorem: every quadratic-like map is hybrid equivalent to a unique f_c(z) = z² + c. Straightening map χ: c ↦ parameter of equivalent quadratic. For quadratic-like, χ continuous ⇒ M contains infinitely many small copies of itself (renormalization). Explains self-similarity of M. Douady rabbit: filled Julia set with threefold symmetry.
Source: Wikipedia; Douady & Hubbard (1985)
Dependency Flowchart
graph TD
DH["Douady-Hubbard Connected"]
Quad["Quadratic Julia & Mandelbrot"]
DefPL["Def: Polynomial-like f: U→V\nU ⋐ V, proper, deg d"]
DefQuadLike["Def: Quadratic-like\ndegree 2"]
DefHybrid["Def: Hybrid equivalence\nQC conjugate, conformal on K"]
ThmStraight["Thm: Straightening\nQuad-like ~ unique f_c"]
DefChi["Def: Straightening map χ\nc = χ(f)"]
ThmCont["Thm: χ continuous\nFor quadratic-like"]
ThmSmall["Thm: M has small copies\nInfinitely many mini-Mandelbrots"]
DH --> DefPL
Quad --> DefPL
DefPL --> DefQuadLike
DefQuadLike --> DefHybrid
DefHybrid --> ThmStraight
ThmStraight --> DefChi
DefChi --> ThmCont
ThmCont --> ThmSmall
classDef axiom fill:#ff6b6b,color:#fff,stroke:#c0392b
classDef definition fill:#b197fc,color:#fff,stroke:#9775fa
classDef theorem fill:#51cf66,color:#fff,stroke:#40c057
class DH,Quad axiom
class DefPL,DefQuadLike,DefHybrid,DefChi definition
class ThmStraight,ThmCont,ThmSmall theorem
Color Scheme
Red Prerequisites
Blue Definitions
Teal Theorems
Process Statistics
- Nodes: 10
- Edges: 11
- Axioms: 2
- Definitions: 4
- Theorems: 3