| import OAI.Dynamics.StandardMap.BandFlux |
|
|
| namespace OAI |
|
|
| open MeasureTheory Set |
| open scoped ENNReal BigOperators |
|
|
| open Set Filter MeasureTheory Topology |
| open scoped ENNReal Classical |
| namespace StandardMapEntropy |
| lemma lengthBand_disjoint : Pairwise (fun i j : ℤ => Disjoint (lengthBand i) (lengthBand j)) := by |
| intro i j hij |
| rw [disjoint_left] |
| intro d hi hj |
| have hpow : StrictMono (fun n:ℤ => (2:ℝ)^n) := zpow_right_strictMono₀ (by norm_num) |
| rcases lt_or_gt_of_ne hij with h|h |
| · have hh := hpow.monotone (by omega : i + 1 ≤ j) |
| exact (not_le_of_gt hi.2.2.2) (hh.trans hj.2.2.1) |
| · have hh := hpow.monotone (by omega : j + 1 ≤ i) |
| exact (not_le_of_gt hj.2.2.2) (hh.trans hi.2.2.1) |
| lemma mem_union_lengthBand {d:NonAffineArray} (hu:UnitArray d.val) (hd:d.val∈twoRayClass) (hl:0<coreLength d.val) : d∈⋃j:ℤ,lengthBand (-j) := by |
| obtain ⟨j,hj⟩ := exists_mem_Ico_zpow hl (by norm_num : (1:ℝ)<2) |
| exact mem_iUnion.mpr ⟨-j,by simpa only [neg_neg,lengthBand,mem_Ico,mem_ofPred_eq] using And.intro hu (And.intro hd hj)⟩ |
| lemma restrict_lengthBand (μ:Measure NonAffineArray) (hu:∀ᵐd ∂μ,UnitArray d.val) (hS:μ.map nonaffineDilate=μ) (j:ℤ) : |
| μ.restrict (lengthBand (-j))=(bandLaw μ).map (scaleZ j) := by |
| have hh := Measure.restrict_map (μ:=μ) (measurable_scaleZ j) (measurableSet_lengthBand (-j)) |
| rw [map_scaleZ μ hu hS j,lengthBand_scaleZ,add_neg_cancel] at hh |
| exact hh |
| lemma bandSeries_law (μ:Measure NonAffineArray) (hu:∀ᵐd ∂μ,UnitArray d.val) (hS:μ.map nonaffineDilate=μ) : |
| μ.restrict (⋃j:ℤ,lengthBand (-j))=Measure.sum (fun j:ℤ => (bandLaw μ).map (scaleZ j)) := by |
| rw [Measure.restrict_iUnion (fun i j hij => lengthBand_disjoint (by omega : -i ≠ -j)) (fun j => measurableSet_lengthBand (-j))] |
| congr 1; funext j; exact restrict_lengthBand μ hu hS j |
| lemma band_integral_series (μ:Measure NonAffineArray) (hu:∀ᵐd ∂μ,UnitArray d.val) (hS:μ.map nonaffineDilate=μ) |
| (g:NonAffineArray→ℝ) (hg:Integrable g μ) : |
| HasSum (fun j:ℤ => ∫d,g (scaleZ j d) ∂bandLaw μ) (∫d in ⋃j:ℤ,lengthBand (-j),g d ∂μ) := by |
| have hh : Integrable g (Measure.sum (fun j:ℤ => (bandLaw μ).map (scaleZ j))) := by rw [←bandSeries_law μ hu hS]; exact hg.restrict |
| have hm (j:ℤ) : Integrable g ((bandLaw μ).map (scaleZ j)) := hh.mono_measure (Measure.le_sum (fun index : ℤ => (bandLaw μ).map (scaleZ index)) j) |
| have he (j:ℤ) : (∫d,g d ∂(bandLaw μ).map (scaleZ j))=∫d,g (scaleZ j d) ∂bandLaw μ := |
| integral_map (measurable_scaleZ j).aemeasurable (hm j).aestronglyMeasurable |
| have hh' := hasSum_integral_measure hh |
| simp_rw [he] at hh' |
| rwa [←bandSeries_law μ hu hS] at hh' |
| lemma bilateral_coboundary_nonpos (F:ℤ→ℝ) (A:ℝ) (h:HasSum (fun j:ℤ => F (j+1)-F j) A) |
| (hf:∀ᶠn:ℕ in atTop,F (n:ℤ)≤F (-(n:ℤ))) : A≤0 := by |
| have ht : Tendsto (fun n:ℕ => ∑j∈Finset.Ico (-(n:ℤ)) (n:ℤ),(F (j+1)-F j)) atTop (𝓝 A) := |
| h.comp Finset.tendsto_Ico_neg |
| have he (n:ℕ) : (∑j∈Finset.Ico (-(n:ℤ)) (n:ℤ),(F (j+1)-F j))=F (n:ℤ)-F (-(n:ℤ)) := by |
| have hh := Finset.sum_Ico_int_sub n (fun j => -F j) |
| simpa only [neg_sub_neg] using hh |
| simp_rw [he] at ht |
| exact le_of_tendsto ht (hf.mono (fun n hn => sub_nonpos.mpr hn)) |
| end StandardMapEntropy |
| |
| end OAI |
| |
