| import Mathlib |
| import OAI.Analysis.Conductivity.Fourier.AngularFourierConjugacy |
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| namespace OAI |
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| section |
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| noncomputable section |
| namespace ScalarConductivity |
| open Set Filter Topology MeasureTheory Matrix |
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| def angularRealTranslation (v : Fin 2 → ℝ) (x : Coord3) : Coord3 := |
| ![x 0,x 1+v 0,x 2+v 1] |
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| def angularShiftPhase (phase : (Fin 2 → ℤ) → ℝ) (v : Fin 2 → ℝ) |
| (h : Fin 2 → ℤ) : ℝ := phase h+(h 0:ℝ)*v 0+(h 1:ℝ)*v 1 |
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| lemma flatPhaseMode_angular_translation (s : Fin 3 → ℝ) (h : Fin 2 → ℤ) |
| (phase : ℝ) (v : Fin 2 → ℝ) (x : Coord3) : |
| flatPhaseMode s h phase (angularRealTranslation v x)= |
| flatPhaseMode s h (phase+(h 0:ℝ)*v 0+(h 1:ℝ)*v 1) x := by |
| unfold flatPhaseMode |
| congr 1 |
| congr 1 |
| simp only [torusAngular,_root_.add_apply,_root_.smul_apply, |
| ContinuousLinearMap.proj_apply,smul_eq_mul] |
| dsimp [angularRealTranslation] |
| ring |
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| theorem flatFourier_angular_translation (s : Fin 3 → ℝ) |
| (a phase : (Fin 2 → ℤ) → ℝ) (v : Fin 2 → ℝ) (x : Coord3) : |
| flatFourier s a phase (angularRealTranslation v x)= |
| flatFourier s a (angularShiftPhase phase v) x := by |
| unfold flatFourier |
| apply tsum_congr |
| intro h |
| rw [flatPhaseMode_angular_translation] |
| rfl |
|
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| lemma alignedPhaseTranslation_zero {k : ℤ} (hk : 0<k) (phase : ℝ) : |
| phase+((![0,k] : Fin 2 → ℤ) 0:ℝ)*((![0,-phase/(k:ℝ)] : Fin 2 → ℝ) 0)+ |
| ((![0,k] : Fin 2 → ℤ) 1:ℝ)*((![0,-phase/(k:ℝ)] : Fin 2 → ℝ) 1)=0 := by |
| have hkR : (k:ℝ)≠0 := by exact_mod_cast hk.ne' |
| simp only [cons_val_zero,cons_val_one,Int.cast_zero,zero_mul,add_zero] |
| field_simp |
| ring |
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| theorem leading_mode_normalized_alignment {s : Fin 3 → ℝ} |
| (hs : ∀ x y : ℝ,(1/2)*(x^2+y^2) ≤ s 0*x^2+2*s 1*x*y+s 2*y^2) |
| {h : Fin 2 → ℤ} (hh : h≠0) (phase : ℝ) : |
| ∃ (k : ℤ) (A : Matrix (Fin 2) (Fin 2) ℤ) (v : Fin 2 → ℝ), |
| 0<k ∧ A.det=1 ∧ h ᵥ* A=![0,k] ∧ |
| (∀ x y : ℝ,(1/2)*(x^2+y^2) ≤ normalizedAngularTensor s A 0*x^2+ |
| 2*normalizedAngularTensor s A 1*x*y+normalizedAngularTensor s A 2*y^2) ∧ |
| (∀ x,flatPhaseMode s h phase (angularNormalizedLift A (angularRealTranslation v x))= |
| flatPhaseMode (normalizedAngularTensor s A) ![0,k] 0 x) ∧ |
| MeasurePreserving (fun x : UnitAddTorus (Fin 2) => angularIntMap A (x+ |
| fun i => ((v i/(2*Real.pi):ℝ) : UnitAddCircle))) volume volume := by |
| obtain ⟨k,A,hk,hA,he⟩ := leading_frequency_SL2_alignment hh |
| let v : Fin 2 → ℝ := ![0,-phase/(k:ℝ)] |
| refine ⟨k,A,v,hk,hA,he,normalizedAngularTensor_lower hs hA,?_,?_⟩ |
| · intro x |
| rw [flatPhaseMode_angular_conjugacy s hA,he,flatPhaseMode_angular_translation] |
| rw [show phase+((![0,k] : Fin 2 → ℤ) 0:ℝ)*v 0+ |
| ((![0,k] : Fin 2 → ℤ) 1:ℝ)*v 1=0 from alignedPhaseTranslation_zero hk phase] |
| · exact (angularIntMap_measurePreserving hA).comp (measurePreserving_add_right volume _) |
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| end ScalarConductivity |
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| end |
| end |
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| end OAI |
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