| import OAI.Analysis.CircleDomains.Uniformization.FiniteCoreCorrespondence | |
| namespace OAI | |
| /-! The compact boundary correspondence is retained by every finite | |
| composition starting from an analytic Jordan configuration. -/ | |
| open Set Function Filter Metric | |
| open scoped Topology OnePoint | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| open FiniteTransfer | |
| namespace FiniteJordanConfiguration | |
| variable {I:Type*} [Finite I] | |
| lemma iterate_analytic_of_initial (C:FiniteJordanConfiguration I) | |
| (ha:∀i,IsAnalyticJordanCap (C.hole i)) (schedule:ℕ→I) (n:ℕ) : | |
| ∀i,IsAnalyticJordanCap ((iterate C schedule n).hole i) := by | |
| induction n with | |
| | zero => exact ha | |
| | succ n ih => | |
| intro i | |
| by_cases he : i=schedule n | |
| · subst i | |
| exact (chosenStep (iterate C schedule n) (schedule n)).analytic_selected | |
| · exact (chosenStep (iterate C schedule n) (schedule n)).analytic_other he (ih i) | |
| lemma exists_cumulative_core_correspondence (C:FiniteJordanConfiguration I) | |
| (ha:∀i,IsAnalyticJordanCap (C.hole i)) (schedule:ℕ→I) (n:ℕ) : | |
| ∃e : closure C.region ≃ₜ closure (iterate C schedule n).region, | |
| ∀p : closure C.region, (p:Sphere)∈C.region → | |
| (e p:Sphere)=cumulativeMap C schedule n p := by | |
| induction n with | |
| | zero => exact ⟨Homeomorph.refl _,fun _ _ => rfl⟩ | |
| | succ n ih => | |
| obtain ⟨e,he⟩ := ih | |
| let q := chosenStep (iterate C schedule n) (schedule n) | |
| obtain ⟨d,hd⟩ := q.exists_core_correspondence | |
| (iterate_analytic_of_initial C ha schedule n (schedule n)) | |
| refine ⟨e.trans d,?_⟩ | |
| intro p hp | |
| have hep : (e p:Sphere)∈(iterate C schedule n).region := by | |
| rw [he p hp] | |
| exact (cumulativeMap_conformal C schedule n).bijOn.mapsTo hp | |
| change (d (e p):Sphere)=stageMap C schedule n (cumulativeMap C schedule n p) | |
| rw [hd (e p) hep,he p hp] | |
| rfl | |
| end FiniteJordanConfiguration | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |