| import OAI.Analysis.IrrationalRotation.Spectral.ProductWeightPolar |
| import OAI.Analysis.IrrationalRotation.Analysis.PurePointDiffuse |
| import OAI.Analysis.IrrationalRotation.Analysis.OrbitFiberNull |
| import OAI.Analysis.IrrationalRotation.Product.ProductOrbitFiberNull |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory Filter |
| open scoped ENNReal |
|
|
| namespace Invariant233.ProductModel |
|
|
| theorem polarConjugate_multiplier_hasCountablyAtomicSpectralType |
| (θ : ℕ → ℝ) (f : Base → ℝ) (C : ℝ) (hC : 0 ≤ C) |
| (hf : Measurable f) (hb : ∀ x, ‖(f x : ℂ)‖ ≤ C) |
| (hcount : ∃ R : Set ℝ, R.Countable ∧ ∀ᵐ x ∂baseMeasure, f x ∈ R) |
| (n : ℤ) : |
| HasCountablyAtomicSpectralType |
| (polarConjugate shiftOperator |
| (multiplier θ (fun x => (f x : ℂ)) C hC hf.complex_ofReal hb) n) := by |
| obtain ⟨R, hR, hmem⟩ := hcount |
| let S : Set ℂ := Complex.ofReal '' R |
| have hS : S.Countable := hR.image _ |
| refine ⟨S, hS, ?_⟩ |
| let b : HilbertBasis ℤ ℂ Fiber := |
| HilbertBasis.ofRepr (LinearIsometryEquiv.refl ℂ _) |
| have hbasis (v : Fiber) (k : ℤ) : inner ℂ (b k) v = v k := |
| (b.repr_apply_apply v k).symm |
| have hmeas (k : ℤ) : Measurable (fun x => (f (rotationIterate θ (k - n) x) : ℂ)) := |
| hf.complex_ofReal.comp (rotationIterate_measurePreserving_polar θ (k - n)).measurable |
| have hrange (k : ℤ) : |
| ∀ᵐ x ∂baseMeasure, (f (rotationIterate θ (k - n) x) : ℂ) ∈ S := by |
| filter_upwards [(rotationIterate_measurePreserving_polar θ (k - n)).quasiMeasurePreserving.ae hmem] |
| with x hx |
| exact Set.mem_image_of_mem Complex.ofReal hx |
| apply MultiplicationPurePoint.topologicalClosure_iSup_eigenspace_eq_top_of_basis b |
| (fun k x => (f (rotationIterate θ (k - n) x) : ℂ)) hmeas hS hrange |
| intro v k |
| simpa only [hbasis] using |
| (polarConjugate_multiplier_coordinate_ae θ _ C hC hf.complex_ofReal hb n v).mono |
| (fun x hx => hx k) |
|
|
| theorem coefficient_fiber_null_of_orbit_fiber_null |
| (θ : ℕ → ℝ) (f : Base → ℝ) |
| (hnull : ∀ z : ℤ → ℝ, |
| baseMeasure {x | ∀ n : ℤ, f (rotationIterate θ n x) = z n} = 0) |
| (ev : ℤ → ℂ) (k : ℤ) : |
| baseMeasure {x | ∀ n : ℤ, (f (rotationIterate θ (k - n) x) : ℂ) = ev n} = 0 := by |
| apply measure_mono_null _ (hnull (fun m => (ev (k - m)).re)) |
| intro x hx m |
| have h := congrArg Complex.re (hx (k - m)) |
| simpa only [sub_sub_cancel, Complex.ofReal_re] using h |
|
|
| theorem jointEigenspace_polarConjugates_eq_bot_of_null_fibers |
| (θ : ℕ → ℝ) (f : Base → ℝ) (C : ℝ) (hC : 0 ≤ C) |
| (hf : Measurable f) (hb : ∀ x, ‖(f x : ℂ)‖ ≤ C) |
| (hnull : ∀ z : ℤ → ℝ, |
| baseMeasure {x | ∀ n : ℤ, f (rotationIterate θ n x) = z n} = 0) |
| (ev : ℤ → ℂ) : |
| jointEigenspace (polarConjugate shiftOperator |
| (multiplier θ (fun x => (f x : ℂ)) C hC hf.complex_ofReal hb)) ev = ⊥ := by |
| apply le_antisymm _ bot_le |
| intro v hv |
| have heigen := mem_jointEigenspace.mp hv |
| have hcoord : ∀ n : ℤ, ∀ᵐ x ∂baseMeasure, ∀ k : ℤ, |
| (f (rotationIterate θ (k - n) x) : ℂ) * v x k = ev n * v x k := by |
| intro n |
| filter_upwards [polarConjugate_multiplier_coordinate_ae θ _ C hC |
| hf.complex_ofReal hb n v, Lp.coeFn_smul (ev n) v] with x hx hs k |
| have h := congrArg (fun w : Hilbert => w x k) (heigen n) |
| rw [hx, hs] at h |
| exact h |
| have havoid : ∀ k : ℤ, ∀ᵐ x ∂baseMeasure, |
| ¬ ∀ n : ℤ, (f (rotationIterate θ (k - n) x) : ℂ) = ev n := by |
| intro k |
| exact measure_eq_zero_iff_ae_notMem.mp |
| (coefficient_fiber_null_of_orbit_fiber_null θ f hnull ev k) |
| have hvzero : v = 0 := by |
| apply Lp.ext |
| filter_upwards [ae_all_iff.mpr hcoord, ae_all_iff.mpr havoid, |
| Lp.coeFn_zero Fiber 2 baseMeasure] with x hx hav hz |
| rw [hz] |
| apply lp.ext |
| funext k |
| change v x k = 0 |
| by_contra hvk |
| apply hav k |
| intro n |
| exact mul_right_cancel₀ hvk (hx n k) |
| exact hvzero |
|
|
| theorem fullPolarAlgebra_multiplier_diffuse_of_null_fibers |
| (θ : ℕ → ℝ) (f : Base → ℝ) (C : ℝ) (hC : 0 ≤ C) |
| (hf : Measurable f) (hb : ∀ x, ‖(f x : ℂ)‖ ≤ C) |
| (hcount : ∃ R : Set ℝ, R.Countable ∧ ∀ᵐ x ∂baseMeasure, f x ∈ R) |
| (hnull : ∀ z : ℤ → ℝ, |
| baseMeasure {x | ∀ n : ℤ, f (rotationIterate θ n x) = z n} = 0) : |
| IsDiffuseVonNeumannAlgebra |
| (fullPolarAlgebra shiftOperator |
| (multiplier θ (fun x => (f x : ℂ)) C hC hf.complex_ofReal hb)) := by |
| apply generated_isDiffuse_of_purePoint_no_jointEigenvectors |
| · exact fullPolarAlgebra_multiplier_abelian θ f C hC hf hb |
| · exact polarConjugate_multiplier_hasCountablyAtomicSpectralType θ f C hC hf hb hcount |
| · exact jointEigenspace_polarConjugates_eq_bot_of_null_fibers θ f C hC hf hb hnull |
|
|
| theorem fullPolarAlgebra_weight_diffuse |
| (θ c : ℕ → ℝ) (hadm : AdmissibleAnglesAndCosts θ c) |
| (hdiv : (∑' i, ENNReal.ofReal (c i * θ i)) = ∞) : |
| IsDiffuseVonNeumannAlgebra |
| (fullPolarAlgebra shiftOperator (star shiftOperator * weightedOperator θ c)) := by |
| rw [star_shift_weightedOperator_eq_multiplier] |
| obtain ⟨S, hS, -, hmem⟩ := ae_weight_mem_countable θ c |
| (fun i => ⟨(hadm.1 i).1.le, (hadm.1 i).2.1⟩) |
| (fun i => (hadm.1 i).2.2.le) hadm.2.1 |
| exact fullPolarAlgebra_multiplier_diffuse_of_null_fibers θ (weight θ c) 1 |
| (by norm_num) (measurable_weight θ c) (weight_complex_norm_le_one θ c) |
| ⟨S, hS, hmem⟩ (weight_rotation_joint_fiber_measure_zero θ c hadm hdiv) |
| |
| end Invariant233.ProductModel |
| |
| end |
| |
| end OAI |
| |
