Description
The Sullivan dictionary establishes systematic parallels between Kleinian groups (discrete groups of Möbius maps) and the dynamics of rational maps. Kleinian Γ ↔ iterates fⁿ; limit set Λ(Γ) ↔ Julia set J(f); domain of discontinuity Ω(Γ) ↔ Fatou set F(f). Sullivan extended these with quasiconformal methods: Ahlfors finiteness (Kleinian) ↔ no wandering domains (rational); Bers finiteness ↔ Shishikura's bound. Enables transfer of ideas between the two fields.
Source: Sullivan; Wikipedia
Dependency Flowchart
graph TD
DefKlein["Def: Kleinian group Γ\nDiscrete Möbius on ℂ̂"]
DefRat["Def: Rational map f\nIteration fⁿ"]
DefLimit["Def: Limit set Λ(Γ)\nAccumulation points"]
DefDomain["Def: Domain Ω(Γ)\n= ℂ̂ \\ Λ"]
DefJulia["Def: Julia set J(f)"]
DefFatou["Def: Fatou set F(f)"]
Dict1["Dict: Λ(Γ) ↔ J(f)\nNonempty, no interior"]
Dict2["Dict: Ω(Γ) ↔ F(f)"]
ThmAhlfors["Thm: Ahlfors finiteness\nFinite type surfaces"]
ThmNWD["Thm: No wandering domains\nSullivan"]
Dict3["Dict: Ahlfors ↔ NWD"]
DefKlein --> DefLimit
DefKlein --> DefDomain
DefRat --> DefJulia
DefRat --> DefFatou
DefLimit --> Dict1
DefJulia --> Dict1
DefDomain --> Dict2
DefFatou --> Dict2
ThmAhlfors --> Dict3
ThmNWD --> Dict3
classDef definition fill:#b197fc,color:#fff,stroke:#9775fa
classDef theorem fill:#51cf66,color:#fff,stroke:#40c057
class DefKlein,DefRat,DefLimit,DefDomain,DefJulia,DefFatou definition
class ThmAhlfors,ThmNWD theorem
class Dict1,Dict2,Dict3 definition
Color Scheme
Blue Definitions / Dictionary entries
Teal Theorems
Process Statistics
- Nodes: 12
- Edges: 14
- Definitions: 9
- Theorems: 2