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f1f7953 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | import OAI.Analysis.IrrationalRotation.Model.CharacterLaw
import OAI.Analysis.IrrationalRotation.Sampling.CharacterWeakConvergence
namespace OAI
noncomputable section
namespace Invariant233
namespace CharacterLaws
open MeasureTheory Filter
open scoped Topology
theorem tendsto_characterLaw
(q : ℕ → ℕ → ℕ) (m : ℕ)
(hq : ∀ j i, 0 < q j i) (hmono : ∀ j, Monotone (q j))
(hgap : ∀ i, i + 1 < m → Tendsto
(fun j => (q j (i + 1) : ℝ) / (q j i : ℝ)) atTop atTop) :
Tendsto (fun j => characterLaw (fun i : Fin m => q j i.val)) atTop
(𝓝 (torusHaar m)) := by
apply (ProbabilityMeasure.tendsto_iff_forall_integral_rclike_tendsto ℂ).mpr
intro f
have h := CharacterWeakConvergence.tendsto_integral_joint_characters
q m hq hmono hgap f.toContinuousMap
have hmap (j : ℕ) :
(∫ y, f y ∂(characterLaw (fun i : Fin m => q j i.val) : Measure _)) =
∫ x : AddCircle (1 : ℝ), f (fun i : Fin m => (q j i.val : ℤ) • x) := by
change (∫ y, f y ∂Measure.map (characterMap (fun i : Fin m => q j i.val)) volume) = _
rw [integral_map
(characterMap (fun i : Fin m => q j i.val)).continuous.measurable.aemeasurable
f.continuous.aestronglyMeasurable]
rfl
simpa only [hmap, BoundedContinuousFunction.coe_toContinuousMap,
torusHaar, ProbabilityMeasure.coe_mk] using h
theorem tendsto_apply_of_uniform_bound
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]
(L : ℕ → E →L[ℂ] ℂ) (M : E →L[ℂ] ℂ)
(hL : ∀ j, ‖L j‖ ≤ 1)
(hconv : ∀ f, Tendsto (fun j => L j f) atTop (𝓝 (M f)))
{f : ℕ → E} {g : E} (hf : Tendsto f atTop (𝓝 g)) :
Tendsto (fun j => L j (f j)) atTop (𝓝 (M g)) := by
have hnorm : Tendsto (fun j => ‖f j - g‖) atTop (𝓝 0) := by
have hg : Tendsto (fun _ : ℕ => g) atTop (𝓝 g) := tendsto_const_nhds
simpa using (hf.sub hg).norm
have hsmall : Tendsto (fun j => L j (f j - g)) atTop (𝓝 0) := by
apply tendsto_zero_iff_norm_tendsto_zero.mpr
apply squeeze_zero (fun _ => norm_nonneg _) (fun j => ?_) hnorm
calc
‖L j (f j - g)‖ ≤ ‖L j‖ * ‖f j - g‖ := (L j).le_opNorm _
_ ≤ 1 * ‖f j - g‖ := mul_le_mul_of_nonneg_right (hL j) (norm_nonneg _)
_ = ‖f j - g‖ := one_mul _
simpa only [map_sub, sub_add_cancel, zero_add] using hsmall.add (hconv g)
theorem tendsto_integral_character_phase
{Z : Type*} [TopologicalSpace Z]
(q : ℕ → ℕ → ℕ) (m : ℕ)
(hq : ∀ j i, 0 < q j i) (hmono : ∀ j, Monotone (q j))
(hgap : ∀ i, i + 1 < m → Tendsto
(fun j => (q j (i + 1) : ℝ) / (q j i : ℝ)) atTop atTop)
{a : ℕ → Z} {b : Z} (ha : Tendsto a atTop (𝓝 b))
(f : C(Z × UnitAddTorus (Fin m), ℂ)) :
Tendsto (fun j => ∫ x : AddCircle (1 : ℝ),
f (a j, fun i : Fin m => (q j i.val : ℤ) • x)) atTop
(𝓝 (∫ y : UnitAddTorus (Fin m), f (b, y))) := by
let : IsProbabilityMeasure (volume : Measure (AddCircle (1 : ℝ))) :=
circleHaar.property
let L : ℕ → C(UnitAddTorus (Fin m), ℂ) →L[ℂ] ℂ :=
fun j => CharacterWeakConvergence.pullbackIntegral volume
(characterMap (fun i : Fin m => q j i.val))
let M : C(UnitAddTorus (Fin m), ℂ) →L[ℂ] ℂ :=
CharacterWeakConvergence.pullbackIntegral volume (ContinuousMap.id _)
exact tendsto_apply_of_uniform_bound L M
(fun j => CharacterWeakConvergence.pullbackIntegral_norm_le volume _)
(fun g => CharacterWeakConvergence.tendsto_integral_joint_characters
q m hq hmono hgap g)
(f.curry.continuous.tendsto b |>.comp ha)
theorem integral_characterPhaseLaw {Z : Type*} [TopologicalSpace Z]
[MeasurableSpace Z] [BorelSpace Z] {m : ℕ} (q : Fin m → ℕ) (a : Z)
(f : C(Z × UnitAddTorus (Fin m), ℂ)) :
(∫ z, f z ∂(characterPhaseLaw q a : Measure _)) =
∫ x : AddCircle (1 : ℝ), f (a, characterMap q x) := by
change (∫ z, f z ∂Measure.map (fun x => (a, characterMap q x)) volume) = _
rw [integral_map
(by fun_prop : Continuous (fun x => (a, characterMap q x))).measurable.aemeasurable
f.continuous.aestronglyMeasurable]
theorem integral_haarPhaseLaw {Z : Type*} [TopologicalSpace Z]
[MeasurableSpace Z] [BorelSpace Z] (m : ℕ) (a : Z)
(f : C(Z × UnitAddTorus (Fin m), ℂ)) :
(∫ z, f z ∂(haarPhaseLaw m a : Measure _)) =
∫ y : UnitAddTorus (Fin m), f (a, y) := by
change (∫ z, f z ∂Measure.map (fun y : UnitAddTorus (Fin m) => (a, y)) volume) = _
rw [integral_map
(by fun_prop : Continuous (fun y : UnitAddTorus (Fin m) => (a, y))).measurable.aemeasurable
f.continuous.aestronglyMeasurable]
theorem tendsto_characterPhaseLaw
{Z : Type*} [TopologicalSpace Z] [MeasurableSpace Z] [BorelSpace Z]
(q : ℕ → ℕ → ℕ) (m : ℕ)
(hq : ∀ j i, 0 < q j i) (hmono : ∀ j, Monotone (q j))
(hgap : ∀ i, i + 1 < m → Tendsto
(fun j => (q j (i + 1) : ℝ) / (q j i : ℝ)) atTop atTop)
{a : ℕ → Z} {b : Z} (ha : Tendsto a atTop (𝓝 b)) :
Tendsto (fun j => characterPhaseLaw (fun i : Fin m => q j i.val) (a j)) atTop
(𝓝 (haarPhaseLaw m b)) := by
apply (ProbabilityMeasure.tendsto_iff_forall_integral_rclike_tendsto ℂ).mpr
intro f
have h := tendsto_integral_character_phase q m hq hmono hgap ha f.toContinuousMap
simpa only [← BoundedContinuousFunction.coe_toContinuousMap,
integral_characterPhaseLaw, integral_haarPhaseLaw, characterMap, ContinuousMap.coe_mk] using h
end CharacterLaws
end Invariant233
end
end OAI
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