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8e4d533 | 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 | import Mathlib
import OAI.Geometry.Cannon.Uniformization.EventualRelativeShrink
import OAI.Geometry.Cannon.Conformal.Distinct
import OAI.Geometry.Cannon.Conformal.QuasiMobius
namespace OAI
/-! A homeomorphism obtained as the limit of surjective finite realizations. -/
section
section
namespace CannonRealization
open Set Filter Topology Metric
variable {X Y : Type*} [MetricSpace X] [MetricSpace Y]
[CompactSpace X] [CompactSpace Y] [PreconnectedSpace X] [Nontrivial Y]
theorem surjective_realization_homeomorph_limit
(F : ℕ → C(X,Y)) (hsurj : ∀ n, Function.Surjective (F n))
(hshrink : EventualRelativeShrink F) :
∃ f : X ≃ₜ Y, RelativeShrink f ∧ ∃ φ : ℕ → ℕ,
StrictMono φ ∧ Tendsto (F ∘ φ) atTop (𝓝 (⟨f,f.continuous⟩ : C(X,Y))) := by
obtain ⟨u,v,huv⟩ := exists_pair_ne Y
obtain ⟨p,hp⟩ := hsurj 0 u
obtain ⟨q,hq⟩ := hsurj 0 v
have hpq : p≠q := by intro h; exact huv (hp.symm.trans ((congrArg (F 0) h).trans hq))
obtain ⟨f,φ,hφ,ht⟩ := compact_map_subsequence F (hshrink.equicontinuous hpq)
have hf := hshrink.limit hφ ht
have hfs := surjective_limit hsurj ht
obtain ⟨p',hp'⟩ := hfs u
obtain ⟨q',hq'⟩ := hfs v
have hn : f p'≠f q' := by rwa [hp',hq']
let H := (Equiv.ofBijective f ⟨hf.injective hn,hfs⟩).toHomeomorphOfContinuousClosed
f.continuous f.continuous.isClosedMap
exact ⟨H,hf,φ,hφ,ht⟩
theorem surjective_realization_quasiMobius_limit [PreconnectedSpace Y]
{Q : ℕ} {P : ℝ} (hpack : CannonPacking.PolynomialPacking (X:=X) Q P)
(hP : 0 ≤ P) (hdiam : ∀ x y : X, dist x y ≤ 1)
(F : ℕ → C(X,Y)) (hsurj : ∀ n, Function.Surjective (F n))
(hshrink : EventualRelativeShrink F) :
∃ f : X ≃ₜ Y, CannonQM.QuasiMobius f ∧ ∃ φ : ℕ → ℕ,
StrictMono φ ∧ Tendsto (F ∘ φ) atTop (𝓝 (⟨f,f.continuous⟩ : C(X,Y))) := by
obtain ⟨f,hf,φ,hφ,ht⟩ := surjective_realization_homeomorph_limit F hsurj hshrink
exact ⟨f,hf.quasiMobius f hpack hP hdiam,φ,hφ,ht⟩
end CannonRealization
end
end
end OAI
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