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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 | import OAI.Analysis.IrrationalRotation.Couplings.PrefixHybrid
import OAI.Analysis.IrrationalRotation.Product.ProductCocycle
namespace OAI
noncomputable section
open MeasureTheory Function Set
namespace Invariant233.ProductModel
theorem finite_floor_support_selectedEndpoint
(θ : ℕ → ℝ) (k : ℤ) (x y : Base) (I : Finset ℕ)
(hx : HasFiniteSupport (fun i =>
⌊circleRepresentative (x i) + (k : ℝ) * θ i⌋))
(hy : HasFiniteSupport (fun i =>
⌊circleRepresentative (y i) + (k : ℝ) * θ i⌋)) :
HasFiniteSupport (fun i =>
⌊circleRepresentative (selectedEndpoint I x y i) + (k : ℝ) * θ i⌋) := by
apply (hx.union hy).subset
intro i hi
by_cases hI : i ∈ I
· right
simpa only [Function.mem_support, selectedEndpoint, ite_eq_left hI] using hi
· left
simpa only [Function.mem_support, selectedEndpoint, ite_eq_right hI] using hi
theorem ae_pair_all_int_finite_floor_support
(θ : ℕ → ℝ) (hθ0 : ∀ i, 0 ≤ θ i) (hθ : Summable θ)
(ν : Measure EndpointPair)
(hfst : ν.map Prod.fst ≪ baseMeasure)
(hsnd : ν.map Prod.snd ≪ baseMeasure) :
∀ᵐ xy ∂ν, ∀ k : ℤ,
HasFiniteSupport (fun i =>
⌊circleRepresentative (xy.1 i) + (k : ℝ) * θ i⌋) ∧
HasFiniteSupport (fun i =>
⌊circleRepresentative (xy.2 i) + (k : ℝ) * θ i⌋) := by
have hbase : ∀ᵐ x ∂baseMeasure, ∀ k : ℤ,
HasFiniteSupport (fun i =>
⌊circleRepresentative (x i) + (k : ℝ) * θ i⌋) :=
CocycleSupport.ae_product_all_int_finite_floor_support θ hθ0 hθ
have hx := ae_of_ae_map measurable_fst.aemeasurable (hfst.ae_le hbase)
have hy := ae_of_ae_map measurable_snd.aemeasurable (hsnd.ae_le hbase)
filter_upwards [hx, hy] with xy hx hy
exact fun k => ⟨hx k, hy k⟩
theorem ae_ordered_cocycle_ratio_cap
(θ c : ℕ → ℝ) (hθ0 : ∀ i, 0 ≤ θ i) (hθ : Summable θ)
(hc : ∀ i, 0 ≤ c i) (ν : Measure EndpointPair)
(hfst : ν.map Prod.fst ≪ baseMeasure)
(hsnd : ν.map Prod.snd ≪ baseMeasure)
(horder : ∀ᵐ xy ∂ν, Ordered xy.1 xy.2) :
∀ᵐ xy ∂ν, ∀ k : ℤ,
0 < ProductCocycle.cocycleVersion θ c k xy.2 /
ProductCocycle.cocycleVersion θ c k xy.1 ∧
ProductCocycle.cocycleVersion θ c k xy.2 /
ProductCocycle.cocycleVersion θ c k xy.1 ≤ 1 := by
filter_upwards [ae_pair_all_int_finite_floor_support θ hθ0 hθ ν hfst hsnd,
horder] with xy hs ho
intro k
exact ⟨div_pos (ProductCocycle.cocycleVersion_pos θ c k xy.2)
(ProductCocycle.cocycleVersion_pos θ c k xy.1),
ProductCocycle.cocycleVersion_ratio_le_one θ c k xy.1 xy.2 hc ho
(hs k).1 (hs k).2⟩
theorem ae_all_selected_cocycle_ratio_cap
(θ c : ℕ → ℝ) (hθ0 : ∀ i, 0 ≤ θ i) (hθ : Summable θ)
(hc : ∀ i, 0 ≤ c i) (ν : Measure EndpointPair)
(hfst : ν.map Prod.fst ≪ baseMeasure)
(hsnd : ν.map Prod.snd ≪ baseMeasure)
(horder : ∀ᵐ xy ∂ν, Ordered xy.1 xy.2) :
∀ᵐ xy ∂ν, ∀ (I : Finset ℕ) (k : ℤ),
0 < ProductCocycle.cocycleVersion θ c k (selectedEndpoint I xy.1 xy.2) /
ProductCocycle.cocycleVersion θ c k xy.1 ∧
ProductCocycle.cocycleVersion θ c k (selectedEndpoint I xy.1 xy.2) /
ProductCocycle.cocycleVersion θ c k xy.1 ≤ 1 := by
filter_upwards [ae_pair_all_int_finite_floor_support θ hθ0 hθ ν hfst hsnd,
horder] with xy hs ho
intro I k
exact ⟨div_pos (ProductCocycle.cocycleVersion_pos θ c k _)
(ProductCocycle.cocycleVersion_pos θ c k _),
ProductCocycle.cocycleVersion_ratio_le_one θ c k xy.1
(selectedEndpoint I xy.1 xy.2) hc (ordered_selectedEndpoint ho I).1
(hs k).1 (finite_floor_support_selectedEndpoint θ k xy.1 xy.2 I
(hs k).1 (hs k).2)⟩
theorem measurableSet_all_cocycle_ratio_cap (θ c : ℕ → ℝ) :
MeasurableSet {xy : EndpointPair | ∀ k : ℤ,
0 < ProductCocycle.cocycleVersion θ c k xy.2 /
ProductCocycle.cocycleVersion θ c k xy.1 ∧
ProductCocycle.cocycleVersion θ c k xy.2 /
ProductCocycle.cocycleVersion θ c k xy.1 ≤ 1} := by
simp only [ofPred_forall]
apply MeasurableSet.iInter
intro k
have hm := ((ProductCocycle.measurable_cocycleVersion θ c k).comp measurable_snd).div
((ProductCocycle.measurable_cocycleVersion θ c k).comp measurable_fst)
exact (measurableSet_lt measurable_const hm).inter
(measurableSet_le hm measurable_const)
theorem prefixPairLaw_cocycle_ratio_cap
(θ c : ℕ → ℝ) (hθ0 : ∀ i, 0 ≤ θ i) (hθ : Summable θ)
(hc : ∀ i, 0 ≤ c i) (ν : Measure EndpointPair)
(hfst : ν.map Prod.fst ≪ baseMeasure)
(hsnd : ν.map Prod.snd ≪ baseMeasure)
(horder : ∀ᵐ xy ∂ν, Ordered xy.1 xy.2) (D : ℕ) :
∀ᵐ xy ∂prefixPairLaw ν D, ∀ k : ℤ,
0 < ProductCocycle.cocycleVersion θ c k xy.2 /
ProductCocycle.cocycleVersion θ c k xy.1 ∧
ProductCocycle.cocycleVersion θ c k xy.2 /
ProductCocycle.cocycleVersion θ c k xy.1 ≤ 1 := by
rw [prefixPairLaw, ae_map_iff (measurable_prefixPair D).aemeasurable
(measurableSet_all_cocycle_ratio_cap θ c)]
filter_upwards [ae_all_selected_cocycle_ratio_cap θ c hθ0 hθ hc ν hfst hsnd horder]
with xy hxy
exact hxy (Finset.range D)
end Invariant233.ProductModel
end
end OAI
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