openAI-math-dataset / lean /OAI /Geometry /Cannon /Uniformization /SurjectiveRealizationHomeomorphLimit.lean
| 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 | |