import OAI.Analysis.BackwardIntertwiners.ErgodicTransitivity namespace OAI namespace BackwardIntertwiners open Filter open scoped Topology noncomputable section open Topology MeasureTheory Set open scoped ENNReal def globalTail (h : ℤ) : Submodule ℂ GlobalHilbert := ⨅ j : ℤ, ⨅ _ : j < h, (coordinateL2 j).ker lemma mem_globalTail (h : ℤ) (f : GlobalHilbert) : f ∈ globalTail h ↔ ∀ᵐ x ∂baseMeasure, f x ∈ coordinateTail h := by simp only [globalTail, Submodule.mem_iInf, LinearMap.mem_ker] constructor · intro hf have h (j : ℤ) : ∀ᵐ x ∂baseMeasure, j < h → f x j = 0 := by by_cases hj : j < h · have hh := (Lp.eq_zero_iff_ae_eq_zero.mp (hf j hj)) filter_upwards [coordinateL2_ae j f, hh] with x hx hzero intro _ exact hx.symm.trans hzero · exact Filter.Eventually.of_forall fun _ => fun h => False.elim (hj h) exact ae_all_iff.mpr h · intro hf j hj apply Lp.eq_zero_iff_ae_eq_zero.mpr filter_upwards [hf, coordinateL2_ae j f] with x hx hcoord exact hcoord.trans (hx j hj) lemma globalTail_closed (h : ℤ) : IsClosed (globalTail h : Set GlobalHilbert) := by have he : (globalTail h : Set GlobalHilbert) = ⋂ j : ℤ, ⋂ _ : j < h, {f : GlobalHilbert | coordinateL2 j f = 0} := by ext; simp [globalTail] rw [he] exact isClosed_iInter fun j => isClosed_iInter fun _ => isClosed_eq (coordinateL2 j).continuous continuous_const lemma constantBasis_mem_globalTail {h j : ℤ} (hj : h ≤ j) : constantBasis j ∈ globalTail h := by rw [mem_globalTail] filter_upwards [constantBasis_ae j] with x hx rw [hx] exact single_mem_coordinateTail hj 1 lemma constantBasis_not_mem_globalTail {h j : ℤ} (hj : j < h) : constantBasis j ∉ globalTail h := by intro hm have hh := (mem_globalTail h _).mp hm have hz : ∀ᵐ x ∂baseMeasure, (1:ℂ) = 0 := by filter_upwards [hh, constantBasis_ae j] with x hx he simpa only [he, lp.single_apply_self] using hx j hj exact one_ne_zero hz.exists.choose_spec lemma globalTail_nonzero (h : ℤ) : globalTail h ≠ ⊥ := by intro he have hmem := constantBasis_mem_globalTail (le_refl h) rw [he] at hmem have hm : constantBasis h = 0 := hmem have hn := constantBasis_norm h rw [hm, norm_zero] at hn norm_num at hn lemma globalTail_proper (h : ℤ) : globalTail h ≠ ⊤ := by intro he apply constantBasis_not_mem_globalTail (show h-1 lp.single 2 j (f x j) := by filter_upwards [singleL2_ae j (coordinateL2 j f), coordinateL2_ae j f] with x h₁ h₂ change singleL2 j (coordinateL2 j f) x = _ rw [h₁, h₂] lemma sliceProjection_singleL2 (j : ℤ) (f : ScalarHilbert) : sliceProjection j (singleL2 j f) = singleL2 j f := by change singleL2 j (coordinateL2 j (singleL2 j f)) = _ rw [coordinateL2_singleL2] lemma sliceProjection_idempotent (j : ℤ) : sliceProjection j * sliceProjection j = sliceProjection j := by ext1 f exact sliceProjection_singleL2 j (coordinateL2 j f) lemma sliceProjection_constantBasis (j k : ℤ) : sliceProjection j (constantBasis k) = if j=k then constantBasis k else 0 := by classical apply Lp.ext filter_upwards [sliceProjection_ae j (constantBasis k), constantBasis_ae k, Lp.coeFn_zero (E:=Fiber) (p:=2) (μ:=baseMeasure)] with x hx he hz rw [hx, he] by_cases hh : j=k · simp [hh, he] · simp only [ite_eq_right hh, lp.single_apply_ne _ _ _ hh, lp.single_zero, hz, Pi.zero_apply] lemma sliceProjection_scalar_commute (j : ℤ) (g : Lp ℂ ∞ baseMeasure) : sliceProjection j * scalarMultiplier g = scalarMultiplier g * sliceProjection j := by ext1 f apply Lp.ext filter_upwards [sliceProjection_ae j (scalarMultiplier g f), scalarMultiplier_ae g f, scalarMultiplier_ae g (sliceProjection j f), sliceProjection_ae j f] with x h₁ h₂ h₃ h₄ change sliceProjection j (scalarMultiplier g f) x = scalarMultiplier g (sliceProjection j f) x rw [h₁,h₂,h₃,h₄] exact lp.single_smul (E:=fun _ : ℤ => ℂ) 2 j (g x) (f x j) lemma sliceProjection_zero_not_commute : sliceProjection 0 * forwardOperator ≠ forwardOperator * sliceProjection 0 := by intro he have h := congrArg (fun T : GlobalHilbert →L[ℂ] GlobalHilbert => T (constantBasis 0)) he simp only [mul_apply_eq_comp, forwardOperator_constantBasis_zero, map_smul, sliceProjection_constantBasis] at h norm_num at h have hh := congrArg norm h rw [norm_zero, forwardOperator_constantBasis_zero, norm_smul, constantBasis_norm, mul_one, Complex.norm_real, Real.norm_of_nonneg (Real.exp_pos _).le] at hh exact (Real.exp_ne_zero (-10)) hh.symm lemma sliceProjection_zero_not_scalar : sliceProjection 0 ∉ Set.range (scalarMultiplier (μ:=baseMeasure)) := by rintro ⟨g,hg⟩ have he0 : scalarMultiplier g (constantBasis 0) = constantBasis 0 := by rw [hg, sliceProjection_constantBasis]; simp have he1 : scalarMultiplier g (constantBasis 1) = 0 := by rw [hg, sliceProjection_constantBasis]; norm_num have h₀ := scalarMultiplier_ae g (constantBasis 0) have h₁ := scalarMultiplier_ae g (constantBasis 1) rw [he0] at h₀ rw [he1] at h₁ have hfalse : ∀ᵐ x ∂baseMeasure, False := by filter_upwards [h₀,h₁,constantBasis_ae 0, constantBasis_ae 1, Lp.coeFn_zero (E:=Fiber) (p:=2) (μ:=baseMeasure)] with x hv₀ hv₁ he₀ he₁ hz have hc₀ := congrArg (fun v : Fiber => v 0) hv₀ have hc₁ := congrArg (fun v : Fiber => v 1) hv₁ simp only [he₀, lp.single_apply_self, lp.coeFn_smul, Pi.smul_apply, smul_eq_mul, mul_one] at hc₀ simp only [hz, he₁, lp.single_apply_self, lp.coeFn_smul, Pi.smul_apply, smul_eq_mul, mul_one, lp.coeFn_zero, Pi.zero_apply] at hc₁ exact one_ne_zero (hc₀.trans hc₁.symm) exact hfalse.exists.choose_spec lemma residue_separates {x y : Base} (h : ∀ m : ℕ, residue m x = residue m y) : x = y := by apply PadicInt.ext_of_toZModPow.mp intro m apply ZMod.val_injective exact h m def residueClass (m t : ℕ) : Set Base := {x | residue m x = t} lemma measurableSet_residueClass (m t : ℕ) : MeasurableSet (residueClass m t) := measurableSet_eq_fun (measurable_residue m) measurable_const lemma residueClass_nonnull {m t : ℕ} (ht : t < 2^m) : baseMeasure (residueClass m t) ≠ 0 := by change baseHaar {x : PadicInt 2 | twoResidue m x = t} ≠ 0 have hcont : Continuous (twoResidue m) := continuous_of_discreteTopology.comp (continuous_twoResidue m) have hOpen : IsOpen {x : PadicInt 2 | twoResidue m x = t} := (isOpen_discrete ({t}:Set ℕ)).preimage hcont have hmem : twoResidue m (t : PadicInt 2) = t := by simp only [twoResidue, map_natCast] exact ZMod.val_natCast_of_lt ht exact (hOpen.measure_pos baseHaar ⟨(t : PadicInt 2),hmem⟩).ne' def residueIndicator (m t : ℕ) : ScalarHilbert := indicatorConstLp 2 (measurableSet_residueClass m t) (measure_ne_top baseMeasure _) (1:ℂ) lemma residueIndicator_ae (m t : ℕ) : residueIndicator m t =ᵐ[baseMeasure] (residueClass m t).indicator (fun _ => (1:ℂ)) := indicatorConstLp_coeFn lemma residueIndicator_ne_zero {m t : ℕ} (ht : t < 2^m) : residueIndicator m t ≠ 0 := by intro hz apply residueClass_nonnull ht rw [measure_eq_zero_iff_ae_notMem] have he := Lp.eq_zero_iff_ae_eq_zero.mp hz filter_upwards [he,residueIndicator_ae m t] with x hx hi intro hmem have h10 : (1:ℂ)=0 := by rw [← Set.indicator_of_mem hmem (fun _ => (1:ℂ)), ← hi]; exact hx exact one_ne_zero h10 lemma residueIndicator_orthogonal (m : ℕ) {s t : ℕ} (hst : s ≠ t) : inner ℂ (residueIndicator m s) (residueIndicator m t) = 0 := by rw [L2.inner_def] apply integral_eq_zero_of_ae filter_upwards [residueIndicator_ae m s, residueIndicator_ae m t] with x hs ht rw [hs,ht] by_cases hxs : x ∈ residueClass m s · have hxt : x ∉ residueClass m t := fun hx => hst (hxs.symm.trans hx) simp [Set.indicator_of_notMem hxt] · simp [Set.indicator_of_notMem hxs] lemma residueIndicator_independent (m : ℕ) : LinearIndependent ℂ (fun t : Fin (2^m) => residueIndicator m t) := by apply linearIndependent_of_ne_zero_of_inner_eq_zero · intro t; exact residueIndicator_ne_zero t.isLt · intro s t hst exact residueIndicator_orthogonal m (fun he => hst (Fin.ext he)) lemma scalarHilbert_infinite : ¬FiniteDimensional ℂ ScalarHilbert := by intro hfin let := hfin have hh := (residueIndicator_independent (Module.finrank ℂ ScalarHilbert + 1)).fintype_card_le_finrank simp only [Fintype.card_fin] at hh have hlt := Nat.lt_two_pow_self (n:=Module.finrank ℂ ScalarHilbert + 1) omega lemma forwardOperator_singleL2_zero (f : ScalarHilbert) : forwardOperator (singleL2 0 f) = (Real.exp (-10):ℂ) • singleL2 1 f := by apply Lp.ext filter_upwards [forwardOperator_ae (singleL2 0 f), singleL2_ae 0 f, singleL2_ae 1 f, Lp.coeFn_smul (Real.exp (-10):ℂ) (singleL2 1 f)] with x h₁ h₂ h₃ h₄ rw [h₁,h₂,h₄,Pi.smul_apply,h₃,forwardFiber_single,explicitBeta_zero] exact lp.single_smul (E:=fun _ : ℤ => ℂ) 2 1 (Real.exp (-10):ℂ) (f x) lemma coordinate_forward_single : (coordinateL2 1).comp (forwardOperator.comp (singleL2 0)) = (Real.exp (-10):ℂ) • (1 : ScalarHilbert →L[ℂ] ScalarHilbert) := by ext1 f change coordinateL2 1 (forwardOperator (singleL2 0 f)) = (Real.exp (-10):ℂ) • f rw [forwardOperator_singleL2_zero, map_smul, coordinateL2_singleL2] lemma forwardOperator_not_compact : ¬IsCompactOperator forwardOperator := by intro hcomp have hc : IsCompactOperator ((coordinateL2 1).comp (forwardOperator.comp (singleL2 0))) := (hcomp.comp_clm (singleL2 0)).clm_comp (coordinateL2 1) rw [coordinate_forward_single] at hc have he : (Real.exp (-10):ℂ) ≠ 0 := by exact_mod_cast Real.exp_ne_zero (-10) have hid : IsCompactOperator (id : ScalarHilbert → ScalarHilbert) := (IsCompactOperator.smul_iff₀ he).mp hc exact scalarHilbert_infinite (FiniteDimensional.of_isCompactOperator_id hid) lemma globalHilbert_infinite : ¬FiniteDimensional ℂ GlobalHilbert := by intro hfin let := hfin apply scalarHilbert_infinite apply FiniteDimensional.of_injective (singleL2 0).toLinearMap intro f g he have he' : singleL2 0 f = singleL2 0 g := he simpa only [coordinateL2_singleL2] using congrArg (coordinateL2 0) he' lemma sliceProjection_selfAdjoint (j : ℤ) : IsSelfAdjoint (sliceProjection j) := by apply ContinuousLinearMap.isSelfAdjoint_iff_isSymmetric.mpr intro f g change inner ℂ (sliceProjection j f) g = inner ℂ f (sliceProjection j g) simp only [L2.inner_def] apply integral_congr_ae filter_upwards [sliceProjection_ae j f, sliceProjection_ae j g] with x hf hg rw [hf,hg,lp.inner_single_left,lp.inner_single_right] lemma sliceProjection_isStarProjection (j : ℤ) : IsStarProjection (sliceProjection j) := ⟨sliceProjection_idempotent j, sliceProjection_selfAdjoint j⟩ end end BackwardIntertwiners end OAI