| 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 |
|
|
