| import OAI.Dynamics.StandardMap.EndpointIntensity |
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| namespace OAI |
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| open MeasureTheory Set |
| open scoped ENNReal BigOperators |
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| open Set Filter MeasureTheory Topology |
| open scoped ENNReal Classical |
| namespace StandardMapEntropy |
| noncomputable def bandLaw (μ:Measure NonAffineArray) : Measure NonAffineArray := μ.restrict (lengthBand 0) |
| noncomputable def bandIntensity (μ:Measure NonAffineArray) : Measure ℝ := (bandLaw μ).map (fun d => endpointLeft d.val) |
| lemma bandLaw_translate (μ:Measure NonAffineArray) (ht:∀r:DyadicTime,μ.map (nonaffineTranslation r)=μ) (r:DyadicTime) : (bandLaw μ).map (nonaffineTranslation r)=bandLaw μ := |
| map_restrict_invariant μ _ (nonaffineTranslation r).continuous.measurable (ht r) _ (measurableSet_lengthBand 0) (lengthBand_translate 0 r) |
| lemma bandIntensity_translate (μ:Measure NonAffineArray) (ht:∀r:DyadicTime,μ.map (nonaffineTranslation r)=μ) (r:DyadicTime) : (bandIntensity μ).map (fun x:ℝ => x-(r:ℝ))=bandIntensity μ := by |
| apply push_invariant _ _ (nonaffineTranslation r).continuous.measurable (bandLaw_translate μ ht r) |
| _ (measurable_endpointLeft.comp measurable_subtype_coe) _ (measurable_id.sub measurable_const) |
| exact (ae_restrict_mem (measurableSet_lengthBand 0)).mono (fun d hd => (endpointLeft_translate r d.val hd.2.1.2.1 hd.2.1.1).symm) |
| lemma bandIntensity_finite (μ:Measure NonAffineArray) [IsFiniteMeasureOnCompacts μ] |
| (hs:∀ᵐd ∂μ,SlowShape (realArray d.val) (999/1000)) : bandIntensity μ (Ico 0 1)<∞ := by |
| have hbin (x:DyadicTime) (hx:0≤(x:ℝ)) : |
| μ {d:NonAffineArray | d∈lengthBand 0 ∧ endpointLeft d.val∈Ico (x:ℝ) ((x:ℝ)+1/2)}<∞ := by |
| apply lt_of_le_of_lt (measure_mono_ae ?_) (separated_secants_finite μ (dyadicInt (-1)) 0 |
| (x+dyadicHalf (dyadicInt 1)) (x+dyadicInt 1) (by norm_num) (by simp only [AddSubgroup.coe_add,dyadicHalf_val,dyadicInt_val,Int.cast_one]; linarith) (999/1000) (by norm_num)) |
| filter_upwards [hs] with d hsd |
| rintro ⟨hd,ha⟩ |
| obtain ⟨hab,hl,hr,hslow⟩ := twoRay_properties hd.1 (by norm_num) hsd hd.2.1 |
| have hlen : 1≤endpointRight d.val-endpointLeft d.val := by simpa only [lengthBand,zpow_zero,coreLength] using hd.2.2.1 |
| refine ⟨?_,?_⟩ |
| · have hh := hl (dyadicInt (-1)) 0 (by norm_num; linarith [ha.1]) (by change (0:ℝ)≤endpointLeft d.val; linarith [ha.1]) |
| simpa only [dyadicInt_val,Int.cast_neg,Int.cast_one,ZeroMemClass.coe_zero,sub_neg_eq_add,zero_add,abs_of_pos (by norm_num : (0:ℝ)<1)] using hh |
| · have hp : ((x+dyadicHalf (dyadicInt 1):DyadicTime):ℝ)=(x:ℝ)+1/2 := by simp only [AddSubgroup.coe_add,dyadicHalf_val,dyadicInt_val,Int.cast_one] |
| have hq : ((x+dyadicInt 1:DyadicTime):ℝ)=(x:ℝ)+1 := by simp only [AddSubgroup.coe_add,dyadicInt_val,Int.cast_one] |
| have hh := hslow (x+dyadicHalf (dyadicInt 1)) (x+dyadicInt 1) |
| (by rw [hp]; constructor <;> linarith [ha.1,ha.2]) (by rw [hq]; constructor <;> linarith [ha.1,ha.2]) |
| rw [abs_of_pos (by rw [hp,hq]; linarith)] at hh |
| exact hh |
| have h0 := hbin 0 (by norm_num) |
| have h1 := hbin (dyadicHalf (dyadicInt 1)) (by norm_num) |
| change ((bandLaw μ).map (endpointLeft ∘ Subtype.val)) (Ico 0 1)<∞ |
| rw [Measure.map_apply (measurable_endpointLeft.comp measurable_subtype_coe) measurableSet_Ico,bandLaw,Measure.restrict_apply ((measurable_endpointLeft.comp measurable_subtype_coe) measurableSet_Ico)] |
| apply lt_of_le_of_lt (measure_mono ?_) ((measure_union_le _ _).trans_lt (ENNReal.add_lt_top.mpr ⟨h0,h1⟩)) |
| rintro d ⟨ha,hd⟩ |
| by_cases h: endpointLeft d.val<1/2 |
| · exact Or.inl ⟨hd,by simpa only [mem_Ico,ZeroMemClass.coe_zero,zero_add,Function.comp_apply] using And.intro ha.1 h⟩ |
| · exact Or.inr ⟨hd,by simp only [mem_Ico,dyadicHalf_val,dyadicInt_val,Int.cast_one]; constructor; exact le_of_not_gt h; have ha1 : endpointLeft d.val<1 := ha.2; linarith⟩ |
| lemma stationary_Ico_finite (μ:Measure ℝ) |
| (ht:∀r:DyadicTime,μ.map (fun x:ℝ => x-(r:ℝ))=μ) (hf:μ (Ico 0 1)<∞) (a b:ℝ) : μ (Icc a b)<∞ := by |
| obtain ⟨n,hn⟩ := exists_nat_gt (b-(⌊a⌋:ℤ)) |
| have he := stationary_Ico_nat μ ht ⌊a⌋ n |
| apply lt_of_le_of_lt (measure_mono (t:=Ico (⌊a⌋:ℝ) ((⌊a⌋:ℝ)+n)) ?_) ?_ |
| · exact fun x hx => ⟨(Int.floor_le a).trans hx.1,by linarith [hx.2]⟩ |
| · rw [he]; exact ENNReal.mul_lt_top (by simp) hf |
| lemma bandLaw_endpoint_finite (μ:Measure NonAffineArray) [IsFiniteMeasureOnCompacts μ] |
| (hs:∀ᵐd ∂μ,SlowShape (realArray d.val) (999/1000)) |
| (ht:∀r:DyadicTime,μ.map (nonaffineTranslation r)=μ) (a b:ℝ) : |
| bandLaw μ {d:NonAffineArray | endpointLeft d.val∈Icc a b}<∞ := by |
| have hh := stationary_Ico_finite (bandIntensity μ) (bandIntensity_translate μ ht) (bandIntensity_finite μ hs) a b |
| change ((bandLaw μ).map (endpointLeft ∘ Subtype.val)) (Icc a b)<∞ at hh |
| rw [Measure.map_apply (measurable_endpointLeft.comp measurable_subtype_coe) measurableSet_Icc] at hh |
| exact hh |
| end StandardMapEntropy |
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| end OAI |
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