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9655094 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 | 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
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