Description
Iteration of rational maps f: ℂ̂→ℂ̂. Julia set J(f) = closure of repelling periodic points; Fatou set F(f) = ℂ̂ \ J(f) = domain of normality. Fatou components: connected components of F(f). Classification: attracting, parabolic, Siegel disk, Herman ring. Wandering domain: component whose iterates are all disjoint. Montel's theorem: normality criterion. Foundation for Sullivan's no wandering domain theorem.
Source: Wikipedia; Milnor, Dynamics in One Complex Variable
Dependency Flowchart
graph TD
DefRat["Def: Rational map\nf: ℂ̂→ℂ̂, deg≥2"]
DefNorm["Def: Normal family\nEquicontinuous"]
DefJulia["Def: Julia set J(f)\nClosure repelling periodic"]
DefFatou["Def: Fatou set F(f)\nℂ̂ \\ J(f)"]
DefComp["Def: Fatou component\nConnected component of F(f)"]
DefWander["Def: Wandering domain\nIterates disjoint"]
ThmMontel["Thm: Montel\n3 omitted values ⇒ normal"]
ThmClass["Thm: Classification\nAttracting, parabolic, Siegel, Herman"]
LemJNonempty["Lem: J(f) nonempty, no interior"]
DefRat --> DefNorm
DefNorm --> DefJulia
DefJulia --> DefFatou
DefFatou --> DefComp
DefComp --> DefWander
ThmMontel --> DefJulia
DefComp --> ThmClass
DefJulia --> LemJNonempty
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 DefRat,DefNorm,DefJulia,DefFatou,DefComp,DefWander definition
class ThmMontel,ThmClass theorem
class LemJNonempty lemma
Color Scheme
Blue Definitions
Teal Theorems
Purple Lemmas
Process Statistics
- Nodes: 9
- Edges: 10
- Definitions: 6
- Lemmas: 1
- Theorems: 2