| import Mathlib |
| import OAI.Geometry.Cannon.Conformal.SphereTile |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace CannonMobius |
| open Set |
| open scoped OnePoint |
|
|
| lemma analytic_fix_infty_eq_affine (f : Sphere ≃ₜ Sphere) (hInfinity : f ∞=∞) |
| (hf : f ∈ biholomorphic) : ∃ a b : ℂ, ∃ ha : a ≠ 0, f=affine a b ha := by |
| obtain ⟨a,b,ha,he⟩ := analytic_fix_infty_affine f hInfinity hf.1 hf.2 |
| refine ⟨a,b,ha,?_⟩ |
| apply Homeomorph.ext |
| intro p |
| induction p using OnePoint.rec with |
| | infty => exact hInfinity |
| | coe z => exact he z |
|
|
| theorem biholomorphic_normal_form (f : Sphere ≃ₜ Sphere) (hf : f ∈ biholomorphic) : |
| (∃ a b : ℂ, ∃ ha : a ≠ 0, f=affine a b ha) ∨ |
| (∃ p a b : ℂ, ∃ ha : a ≠ 0, |
| f=affine 1 p one_ne_zero*inversion*affine a b ha) := by |
| by_cases hInfinity : f ∞=∞ |
| · exact Or.inl (analytic_fix_infty_eq_affine f hInfinity hf) |
| · have hp : ∃ p : ℂ, f ∞=(p:Sphere) := by |
| induction h : f ∞ using OnePoint.rec with |
| | infty => exact (hInfinity h).elim |
| | coe p => exact ⟨p,rfl⟩ |
| obtain ⟨p,hp⟩ := hp |
| let N : Sphere ≃ₜ Sphere := inversion*affine 1 (-p) one_ne_zero |
| have hNg : N ∈ biholomorphic := biholomorphic.mul_mem inversion_biholomorphic |
| (affine_biholomorphic 1 (-p) one_ne_zero) |
| have hNf : (N*f) ∞=∞ := by |
| change inv (affine 1 (-p) one_ne_zero (f ∞))=∞ |
| rw [hp,affine_coe] |
| simp only [one_mul,add_neg_cancel,inv_zero] |
| obtain ⟨a,b,ha,he⟩ := analytic_fix_infty_eq_affine (N*f) hNf (biholomorphic.mul_mem hNg hf) |
| have hNi : N⁻¹=affine 1 p one_ne_zero*inversion := by |
| rw [show N=inversion*affine 1 (-p) one_ne_zero from rfl,mul_inv_rev] |
| change (affine 1 (-p) one_ne_zero).symm*inversion=affine 1 p one_ne_zero*inversion |
| rw [affine_symm] |
| simp only [inv_one,neg_neg,div_one] |
| refine Or.inr ⟨p,a,b,ha,?_⟩ |
| calc |
| f=N⁻¹*(N*f) := by rw [← mul_assoc,inv_mul_cancel,one_mul] |
| _=affine 1 p one_ne_zero*inversion*affine a b ha := by rw [hNi,he] |
|
|
| theorem biholomorphic_mem_of_generators (S : Subgroup (Sphere ≃ₜ Sphere)) |
| (ha : ∀ a b : ℂ, ∀ h : a ≠ 0, affine a b h ∈ S) (hi : inversion ∈ S) |
| {f : Sphere ≃ₜ Sphere} (hf : f ∈ biholomorphic) : f ∈ S := by |
| rcases biholomorphic_normal_form f hf with ⟨a,b,h,rfl⟩ | ⟨p,a,b,h,rfl⟩ |
| · exact ha a b h |
| · exact S.mul_mem (S.mul_mem (ha 1 p one_ne_zero) hi) (ha a b h) |
| end CannonMobius |
|
|
| end |
| end |
|
|
| end OAI |
|
|
