openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Sampling /ErgodicCharacterCriterion.lean
| import Mathlib | |
| namespace OAI | |
| open MeasureTheory Set Filter | |
| open scoped Topology ENNReal | |
| namespace Invariant233.Characters | |
| theorem integral_eq_zero_of_eigenfunction {X : Type*} [MeasurableSpace X] | |
| {μ : Measure X} {σ : X → X} (hσ : MeasurePreserving σ μ μ) | |
| {f : X → ℂ} (hf : AEStronglyMeasurable f μ) {z : ℂ} (hz : z ≠ 1) | |
| (heigen : ∀ᵐ x ∂μ, f (σ x) = z * f x) : | |
| (∫ x, f x ∂μ) = 0 := by | |
| have hfm : AEStronglyMeasurable f (μ.map σ) := by rwa [hσ.map_eq] | |
| have hmap := integral_map hσ.measurable.aemeasurable hfm | |
| rw [hσ.map_eq] at hmap | |
| have heq : (∫ x, f x ∂μ) = z * ∫ x, f x ∂μ := by | |
| calc | |
| _ = ∫ x, f (σ x) ∂μ := hmap | |
| _ = ∫ x, z * f x ∂μ := integral_congr_ae heigen | |
| _ = _ := integral_const_mul z f | |
| have hzero : (1 - z) * (∫ x, f x ∂μ) = 0 := by | |
| calc | |
| _ = (∫ x, f x ∂μ) - z * (∫ x, f x ∂μ) := by ring | |
| _ = 0 := sub_eq_zero.mpr heq | |
| exact (mul_eq_zero.mp hzero).resolve_left (sub_ne_zero.mpr hz.symm) | |
| theorem eq_zero_of_dense_inner {E ι : Type*} [NormedAddCommGroup E] | |
| [InnerProductSpace ℂ E] (v : E) (χ : ι → E) | |
| (hdense : (Submodule.span ℂ (Set.range χ)).topologicalClosure = ⊤) | |
| (hinner : ∀ i, inner ℂ v (χ i) = 0) : v = 0 := by | |
| let L : E →L[ℂ] ℂ := innerSL ℂ v | |
| have hspan : Submodule.span ℂ (Set.range χ) ≤ L.toLinearMap.ker := by | |
| apply Submodule.span_le.mpr | |
| rintro _ ⟨i, rfl⟩ | |
| exact hinner i | |
| have htop := (Submodule.span ℂ (Set.range χ)).topologicalClosure_minimal | |
| hspan L.isClosed_ker | |
| rw [hdense] at htop | |
| have hv : L v = 0 := htop (Submodule.mem_top : v ∈ (⊤ : Submodule ℂ E)) | |
| exact inner_self_eq_zero.mp hv | |
| theorem ergodic_of_dense_L2_eigenfunctions {X ι : Type*} [MeasurableSpace X] | |
| {μ : Measure X} [IsProbabilityMeasure μ] {σ : X → X} | |
| (hσ : MeasurePreserving σ μ μ) (i₀ : ι) (χ : ι → X → ℂ) (z : ι → ℂ) | |
| (hχ : ∀ i, MemLp (χ i) 2 μ) | |
| (hconst : χ i₀ =ᵐ[μ] fun _ => 1) | |
| (hdense : (Submodule.span ℂ | |
| (Set.range (fun i => (hχ i).toLp (χ i)))).topologicalClosure = ⊤) | |
| (hz : ∀ i, i ≠ i₀ → z i ≠ 1) | |
| (heigen : ∀ i, i ≠ i₀ → ∀ᵐ x ∂μ, χ i (σ x) = z i * χ i x) : | |
| Ergodic σ μ := by | |
| classical | |
| refine ⟨hσ, ⟨?_⟩⟩ | |
| intro s hs hinv | |
| let p : ℝ := μ.real s | |
| let u : Lp ℂ 2 μ := indicatorConstLp 2 hs (measure_ne_top μ s) 1 | |
| let e : Lp ℂ 2 μ := indicatorConstLp 2 MeasurableSet.univ (measure_ne_top μ univ) 1 | |
| have hσs : MeasurePreserving σ (μ.restrict s) (μ.restrict s) := by | |
| simpa only [hinv] using hσ.restrict_preimage hs | |
| have hpair : ∀ i, | |
| inner ℂ u ((hχ i).toLp (χ i)) = (p : ℂ) * inner ℂ e ((hχ i).toLp (χ i)) := by | |
| intro i | |
| have hfull : inner ℂ e ((hχ i).toLp (χ i)) = ∫ x, χ i x ∂μ := by | |
| rw [L2.inner_indicatorConstLp_one, setIntegral_univ] | |
| exact integral_congr_ae (hχ i).coeFn_toLp | |
| have hset : inner ℂ u ((hχ i).toLp (χ i)) = ∫ x in s, χ i x ∂μ := by | |
| rw [L2.inner_indicatorConstLp_one] | |
| exact integral_congr_ae ((hχ i).coeFn_toLp.filter_mono (ae_mono Measure.restrict_le_self)) | |
| rw [hfull, hset] | |
| by_cases hi : i = i₀ | |
| · subst i | |
| rw [integral_congr_ae hconst, | |
| integral_congr_ae (hconst.filter_mono (ae_mono Measure.restrict_le_self))] | |
| simp [p] | |
| · have hzero := integral_eq_zero_of_eigenfunction hσ (hχ i).aestronglyMeasurable | |
| (hz i hi) (heigen i hi) | |
| have hzeros := integral_eq_zero_of_eigenfunction hσs | |
| (hχ i).aestronglyMeasurable.restrict (hz i hi) | |
| ((heigen i hi).filter_mono (ae_mono Measure.restrict_le_self)) | |
| rw [hzero, hzeros, mul_zero] | |
| have heq : u = (p : ℂ) • e := by | |
| apply sub_eq_zero.mp | |
| apply eq_zero_of_dense_inner (u - (p : ℂ) • e) | |
| (fun i => (hχ i).toLp (χ i)) hdense | |
| intro i | |
| rw [inner_sub_left, inner_smul_left, hpair i] | |
| simp | |
| have huu : inner ℂ u u = (p : ℂ) := by | |
| change inner ℂ (indicatorConstLp 2 hs (measure_ne_top μ s) (1 : ℂ)) | |
| (indicatorConstLp 2 hs (measure_ne_top μ s) (1 : ℂ)) = _ | |
| rw [L2.inner_indicatorConstLp_indicatorConstLp] | |
| simp [p] | |
| have heu : inner ℂ e u = (p : ℂ) := by | |
| change inner ℂ (indicatorConstLp 2 MeasurableSet.univ (measure_ne_top μ univ) (1 : ℂ)) | |
| (indicatorConstLp 2 hs (measure_ne_top μ s) (1 : ℂ)) = _ | |
| rw [L2.inner_indicatorConstLp_indicatorConstLp] | |
| simp [p] | |
| have hpalg := congrArg (fun v : Lp ℂ 2 μ => inner ℂ v u) heq | |
| rw [huu, inner_smul_left, heu] at hpalg | |
| have hpalg' := congrArg Complex.re hpalg | |
| simp at hpalg' | |
| have hp01 : p = 0 ∨ p = 1 := by | |
| have hpzero : p * (p - 1) = 0 := by nlinarith | |
| rcases mul_eq_zero.mp hpzero with hp | hp | |
| · exact Or.inl hp | |
| · exact Or.inr (sub_eq_zero.mp hp) | |
| apply eventuallyEmptyOrUniv_iff.mpr | |
| rcases hp01 with hp | hp | |
| · right | |
| rw [ae_iff] | |
| simpa only [not_not, Set.ofPred_mem_eq] using (measureReal_eq_zero_iff (μ := μ)).mp hp | |
| · left | |
| rw [ae_iff] | |
| have hpc : μ.real sᶜ = 0 := by rw [measureReal_compl hs, probReal_univ]; exact sub_eq_zero.mpr hp.symm | |
| exact (measureReal_eq_zero_iff (μ := μ)).mp hpc | |
| theorem ergodic_of_dense_continuous_eigenfunctions {X ι : Type*} | |
| [TopologicalSpace X] [T2Space X] [CompactSpace X] [MeasurableSpace X] [BorelSpace X] | |
| {μ : Measure X} [IsProbabilityMeasure μ] [μ.WeaklyRegular] | |
| {σ : X → X} (hσ : MeasurePreserving σ μ μ) | |
| (i₀ : ι) (χ : ι → C(X, ℂ)) (z : ι → ℂ) | |
| (hconst : χ i₀ = 1) | |
| (hdense : (Submodule.span ℂ (Set.range χ)).topologicalClosure = ⊤) | |
| (hz : ∀ i, i ≠ i₀ → z i ≠ 1) | |
| (heigen : ∀ i, i ≠ i₀ → ∀ x, χ i (σ x) = z i * χ i x) : | |
| Ergodic σ μ := by | |
| let hχ : ∀ i, MemLp (χ i) 2 μ := fun i => ContinuousMap.memLp μ ℂ (χ i) | |
| have hlp : (fun i => (hχ i).toLp (χ i)) = | |
| (fun i => ContinuousMap.toLp 2 μ ℂ (χ i)) := by | |
| funext i | |
| apply Lp.ext | |
| exact (hχ i).coeFn_toLp.trans (ContinuousMap.coeFn_toLp μ (χ i)).symm | |
| have hdenseLp : (Submodule.span ℂ | |
| (Set.range (fun i => (hχ i).toLp (χ i)))).topologicalClosure = ⊤ := by | |
| rw [hlp] | |
| have hd := (ContinuousMap.toLp_denseRange ℂ μ ℂ (by norm_num : (2 : ℝ≥0∞) ≠ ∞)).topologicalClosure_map_submodule hdense | |
| rw [Submodule.map_span, ← Set.range_comp'] at hd | |
| exact hd | |
| refine ergodic_of_dense_L2_eigenfunctions hσ i₀ (fun i => χ i) z hχ ?_ hdenseLp hz ?_ | |
| · exact Eventually.of_forall fun x => by rw [hconst]; rfl | |
| · intro i hi | |
| exact Eventually.of_forall (heigen i hi) | |
| end Invariant233.Characters | |
| end OAI | |