File size: 5,553 Bytes
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 | import OAI.Analysis.IrrationalRotation.Model.CircleRationalGraphs
import OAI.Analysis.IrrationalRotation.Rotation.CircleRationalGraphCaps
import OAI.Analysis.IrrationalRotation.Operators.OperatorCompactTransfer
import OAI.Analysis.IrrationalRotation.Analysis.CountableGraphTransfer
import OAI.Analysis.IrrationalRotation.Analysis.UniformAbsoluteContinuity
namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped Topology
namespace Invariant233.CircleParameterLaws
variable {θ : ℝ} {p : CircleParameterData} (hp : CircleParameterLaws θ p)
include hp
theorem rational_graph_measure_ratio_cap (A : ℕ) (hA : 1 ≤ A) (r k : ℤ) :
∀ᵐ z ∂(circleTranslationGraph (p.rationalTranslation A r) : Measure (Circle × Circle)),
p.floorCocycle θ k z.2 / p.floorCocycle θ k z.1 ≤ p.rationalRatioBound A := by
change ∀ᵐ z ∂circleMeasure.map (fun x => (x, x + p.rationalTranslation A r)), _
apply (ae_map_iff (measurable_id.prodMk
(measurable_id.add measurable_const)).aemeasurable ?_).mpr
· filter_upwards [hp.ae_rational_graph_ratio_cap] with x hx
exact hx A hA r k
· exact measurableSet_le
((hp.measurable_parameter_cocycle k).comp measurable_snd |>.div
((hp.measurable_parameter_cocycle k).comp measurable_fst)) measurable_const
theorem rational_graph_identity_of_approximants
(P : Circle → SequenceHilbert ℤ →L[ℂ] SequenceHilbert ℤ)
(hP : ∀ x, IsStarProjection (P x))
(hPm : ∀ k : ℤ, Measurable (fun x => P x (lp.single 2 k 1)))
(A : ℕ) (hA : 1 ≤ A) (r : ℤ)
(ν : ℕ → FiniteMeasure (Circle × Circle))
(hν : Tendsto ν atTop (𝓝 (circleTranslationGraph (p.rationalTranslation A r))))
(hfirst : UniformlyAbsolutelyContinuous circleMeasure
(fun j => (ν j : Measure (Circle × Circle)).map Prod.fst))
(hsecond : UniformlyAbsolutelyContinuous circleMeasure
(fun j => (ν j : Measure (Circle × Circle)).map Prod.snd))
(hcap : ∀ j k, ∀ᵐ z ∂(ν j : Measure (Circle × Circle)),
p.floorCocycle θ k z.2 / p.floorCocycle θ k z.1 ≤ p.rationalRatioBound A)
(hzero : ∀ j, ∀ᵐ z ∂(ν j : Measure (Circle × Circle)),
(1 - P z.2).comp ((ratioDiagonal (p.floorCocycle θ) z).comp (P z.1)) = 0) :
∀ᵐ x ∂circleMeasure,
(1 - P (x + p.rationalTranslation A r)) *
ratioDiagonal (p.floorCocycle θ) (x, x + p.rationalTranslation A r) * P x = 0 := by
let : IsFiniteMeasure circleMeasure := by
change IsFiniteMeasure (volume : Measure Circle)
infer_instance
have hlim := CompactTransfer.compact_operator_transfer circleMeasure P hP
(p.floorCocycle θ) (fun k x => p.floorCocycle_pos θ k x)
hp.measurable_parameter_cocycle hPm hν hfirst hsecond
(p.rationalRatioBound A) (Real.exp_pos _)
hcap (fun k => hp.rational_graph_measure_ratio_cap A hA r k) hzero
have h := ae_of_ae_map
(measurable_id.prodMk (measurable_id.add measurable_const)).aemeasurable hlim
filter_upwards [h] with x hx
exact hx
theorem rational_graph_identity_of_orbit_approximants
(P : Circle → SequenceHilbert ℤ →L[ℂ] SequenceHilbert ℤ)
(hP : ∀ x, IsStarProjection (P x))
(hPm : ∀ k : ℤ, Measurable (fun x => P x (lp.single 2 k 1)))
(horbit : ∀ n : ℕ, 0 < n → ∀ᵐ x ∂circleMeasure,
(1 - P (CircleWeights.rotate θ (n : ℤ) x)).comp
((ratioDiagonal (p.floorCocycle θ)
(x, CircleWeights.rotate θ (n : ℤ) x)).comp (P x)) = 0)
(A : ℕ) (hA : 1 ≤ A) (r : ℤ)
(ν : ℕ → FiniteMeasure (Circle × Circle))
(hν : Tendsto ν atTop (𝓝 (circleTranslationGraph (p.rationalTranslation A r))))
(hfirst : UniformlyAbsolutelyContinuous circleMeasure
(fun j => (ν j : Measure (Circle × Circle)).map Prod.fst))
(hsecond : UniformlyAbsolutelyContinuous circleMeasure
(fun j => (ν j : Measure (Circle × Circle)).map Prod.snd))
(hcap : ∀ j k, ∀ᵐ z ∂(ν j : Measure (Circle × Circle)),
p.floorCocycle θ k z.2 / p.floorCocycle θ k z.1 ≤ p.rationalRatioBound A)
(hsupport : ∀ j, ∀ᵐ z ∂(ν j : Measure (Circle × Circle)),
∃ n : ℕ, 0 < n ∧ z.2 = CircleWeights.rotate θ (n : ℤ) z.1) :
∀ᵐ x ∂circleMeasure,
(1 - P (x + p.rationalTranslation A r)) *
ratioDiagonal (p.floorCocycle θ) (x, x + p.rationalTranslation A r) * P x = 0 := by
apply hp.rational_graph_identity_of_approximants P hP hPm A hA r ν hν hfirst hsecond hcap
intro j
apply Fields.ae_pair_relation_of_countable_graphs
(fun x y => (1 - P y).comp ((ratioDiagonal (p.floorCocycle θ) (x, y)).comp (P x)) = 0)
(fun n : ℕ+ => CircleWeights.rotate θ (n : ℤ))
(hfirst.absolutelyContinuous j)
· intro n
exact horbit n n.property
· filter_upwards [hsupport j] with z hz
obtain ⟨n, hn, he⟩ := hz
exact ⟨⟨n, hn⟩, he⟩
omit hp in
theorem rationalGraphIdentities_of_forall (_hp : CircleParameterLaws θ p)
(P : Circle → SequenceHilbert ℤ →L[ℂ] SequenceHilbert ℤ)
(h : ∀ A : ℕ, 1 ≤ A → ∀ r : ℤ, ∀ᵐ x ∂circleMeasure,
(1 - P (x + p.rationalTranslation A r)) *
ratioDiagonal (p.floorCocycle θ) (x, x + p.rationalTranslation A r) * P x = 0) :
p.RationalGraphIdentities θ P := by
apply ae_all_iff.mpr
intro A
by_cases hA : 1 ≤ A
· filter_upwards [ae_all_iff.mpr (h A hA)] with x hx
exact fun _ => hx
· exact Filter.Eventually.of_forall fun _ hAx => (hA hAx).elim
end Invariant233.CircleParameterLaws
end
end OAI
|