Description
Sullivan's Theorem (1985): For a rational map f: ℂ̂→ℂ̂ of degree ≥ 2, every Fatou component is eventually periodic. Equivalently: rational maps have no wandering domains. Proof: assume wandering U exists; Baker's lemma ⇒ U eventually simply-connected; deform conformal structure along grand orbit via Beltrami differentials; MRMT yields infinite-dimensional family of conjugate rational maps; contradiction (Rat_d is finite-dimensional).
Source: Wikipedia; Sullivan; Baker–McMullen simplification
Dependency Flowchart
graph TD
Julia["Julia & Fatou sets"]
MRMT["Measurable Riemann Mapping"]
DefWander["Def: Wandering domain\nIterates disjoint"]
DefPeriodic["Def: Eventually periodic\nf^n(U)=f^m(U)"]
LemBaker["Lem: Baker\nWandering ⇒ eventually simply-connected"]
ThmNWD["Thm: Sullivan\nNo wandering domains"]
LemDeform["Lem: QC deformation\nBeltrami along grand orbit"]
LemFinDim["Lem: Rat_d finite-dimensional\nZariski open in ℂP^{2d+1}"]
Julia --> DefWander
Julia --> DefPeriodic
DefWander --> LemBaker
LemBaker --> LemDeform
MRMT --> LemDeform
LemDeform --> ThmNWD
LemFinDim --> ThmNWD
DefPeriodic --> ThmNWD
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 Julia,MRMT axiom
class DefWander,DefPeriodic definition
class ThmNWD theorem
class LemBaker,LemDeform,LemFinDim lemma
Color Scheme
Red Prerequisites
Blue Definitions
Teal Theorems
Green Lemmas
Process Statistics
- Nodes: 9
- Edges: 11
- Axioms: 2
- Definitions: 2
- Lemmas: 3
- Theorems: 1