| import OAI.Dynamics.StandardMap.BridgeCritical |
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| namespace OAI |
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| open MeasureTheory Set |
| open scoped ENNReal BigOperators |
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| open MeasureTheory Set Filter Metric |
| open scoped ENNReal Topology CompactlySupported Classical |
| namespace StandardMapEntropy |
| noncomputable def arrayBridgeEvent {Rw Rh : ℕ} (B : BridgeGrid Rw Rh) |
| (Fw : (Fin Rw → ℝ) → ℝ) (Fh : (Fin Rh → ℝ) → ℝ) (ηw ηh : ℝ) : Set DistanceArray := |
| {d | (999/1000:ℝ)*((B.v:ℝ)-(B.u:ℝ))<d.val B.u B.v ∧ |
| ηw<arrayObservation B.sw B.tw Fw d ∧ ηh<arrayObservation B.sh B.th Fh d} |
| lemma isOpen_arrayBridgeEvent {Rw Rh : ℕ} (B : BridgeGrid Rw Rh) |
| (Fw : (Fin Rw → ℝ) → ℝ) (Fh : (Fin Rh → ℝ) → ℝ) (ηw ηh : ℝ) |
| (hw : Continuous Fw) (hh : Continuous Fh) : IsOpen (arrayBridgeEvent B Fw Fh ηw ηh) := |
| (isOpen_lt continuous_const (continuous_arrayEval B.u B.v)).inter |
| ((isOpen_lt continuous_const (continuous_arrayObservation _ _ _ hw)).inter |
| (isOpen_lt continuous_const (continuous_arrayObservation _ _ _ hh))) |
| namespace CriticalScaleSequence |
| lemma bridge_vague_tendsto (S : CriticalScaleSequence) {Rw Rh : ℕ} (B : BridgeGrid Rw Rh) |
| (p len : ℕ → ℕ) (hp : Tendsto p atTop atTop) |
| (hpl : ∀i j,j<len i → p i+j≤S.exponent i) |
| (Fw : (Fin Rw → ℝ) → ℝ) (Fh : (Fin Rh → ℝ) → ℝ) (L : NNReal) |
| (hFw : LipschitzWith L Fw) (hFh : LipschitzWith L Fh) (hFwz : Fw 0=0) (hFhz : Fh 0=0) |
| (hW0 : ∀d,0≤ arrayObservation B.sw B.tw Fw d) (hH0 : ∀d,0≤ arrayObservation B.sh B.th Fh d) |
| (T : ℝ) (hT : ∀ᶠ i in atTop, |
| (∫d,arrayObservation B.sw B.tw Fw d ∂scaleLaw (S.parameter i) (S.positive i).le (p i) (len i) (S.epsilon i))≤T ∨ |
| (∫d,arrayObservation B.sh B.th Fh d ∂scaleLaw (S.parameter i) (S.positive i).le (p i) (len i) (S.epsilon i))≤T) |
| (ηw ηh : ℝ) (hηw : 0<ηw) (hηh : 0<ηh) (g : C_c(NonAffineArray,ℝ)) |
| (hg : ∀d,0≤g d) (hz : ∀d : NonAffineArray,d.val∉arrayBridgeEvent B Fw Fh ηw ηh → g d=0) : |
| Tendsto (fun i => ∫d,g d ∂nonaffinePart (scaleLaw (S.parameter i) (S.positive i).le (p i) (len i) (S.epsilon i))) atTop (𝓝 0) := by |
| let F := arrayTestExtend g |
| obtain ⟨C,hC⟩ := isCompact_univ.exists_bound_of_continuousOn F.continuous.continuousOn |
| have hFz : ∀d,d∉arrayBridgeEvent B Fw Fh ηw ηh → F d=0 := by |
| intro d hd |
| by_cases ha : d∈affineLocus |
| · exact arrayTestExtend_affine g d ha |
| · exact (arrayTestExtend_apply g ⟨d,ha⟩).trans (hz ⟨d,ha⟩ hd) |
| have hF0 : ∀d,0≤F d := by |
| intro d |
| by_cases ha : d∈affineLocus |
| · rw [show F d=0 from arrayTestExtend_affine g d ha] |
| · rw [arrayTestExtend_apply g ⟨d,ha⟩]; exact hg _ |
| have hFB : ∀d,F d≤ max C 0 := by |
| intro d |
| have hb : |F d|≤C := by convert! hC d (mem_univ d) using 1 |
| exact (le_abs_self _).trans (hb.trans (le_max_left _ _)) |
| have hFW : ∀d,F d≤(max C 0/ηw)*arrayObservation B.sw B.tw Fw d := by |
| intro d |
| by_cases hd : d∈arrayBridgeEvent B Fw Fh ηw ηh |
| · have hpos := hd.2.1 |
| have hh := mul_le_mul_of_nonneg_left hpos.le (div_nonneg (le_max_right C 0) hηw.le) |
| rw [div_mul_cancel₀ _ hηw.ne'] at hh |
| exact (hFB d).trans hh |
| · rw [hFz d hd] |
| exact mul_nonneg (div_nonneg (le_max_right _ _) hηw.le) (hW0 d) |
| have hzero : ∀d,¬((999/1000:ℝ)*((B.v:ℝ)-(B.u:ℝ))<d.val B.u B.v ∧ ηh<arrayObservation B.sh B.th Fh d) → F d=0 := by |
| intro d hd |
| apply hFz |
| exact fun h => hd ⟨h.1,h.2.2⟩ |
| have hh := S.bridge_integral_tendsto B p len hp hpl Fw Fh L hFw hFh hFwz hFhz hW0 hH0 T hT |
| ηh hηh F F.continuous hF0 (max C 0/ηw) (div_nonneg (le_max_right _ _) hηw.le) hFW hzero |
| simpa only [F,integral_arrayTestExtend] using hh |
| lemma bridge_null_of_limit (S : CriticalScaleSequence) {Rw Rh : ℕ} (B : BridgeGrid Rw Rh) |
| (p len : ℕ → ℕ) (hp : Tendsto p atTop atTop) |
| (hpl : ∀i j,j<len i → p i+j≤S.exponent i) |
| (ν : Measure NonAffineArray) [ν.Regular] (l : Filter ℕ) [l.NeBot] (hl : l≤atTop) |
| (hconv : ∀g : C_c(NonAffineArray,ℝ),Tendsto (fun i => ∫d,g d ∂nonaffinePart |
| (scaleLaw (S.parameter i) (S.positive i).le (p i) (len i) (S.epsilon i))) l (𝓝 (∫d,g d ∂ν))) |
| (Fw : (Fin Rw → ℝ) → ℝ) (Fh : (Fin Rh → ℝ) → ℝ) (L : NNReal) |
| (hFw : LipschitzWith L Fw) (hFh : LipschitzWith L Fh) (hFwz : Fw 0=0) (hFhz : Fh 0=0) |
| (hW0 : ∀d,0≤arrayObservation B.sw B.tw Fw d) (hH0 : ∀d,0≤arrayObservation B.sh B.th Fh d) |
| (T : ℝ) (hT : ∀ᶠ i in atTop, |
| (∫d,arrayObservation B.sw B.tw Fw d ∂scaleLaw (S.parameter i) (S.positive i).le (p i) (len i) (S.epsilon i))≤T ∨ |
| (∫d,arrayObservation B.sh B.th Fh d ∂scaleLaw (S.parameter i) (S.positive i).le (p i) (len i) (S.epsilon i))≤T) |
| (ηw ηh : ℝ) (hηw : 0<ηw) (hηh : 0<ηh) : |
| ν {d | d.val∈arrayBridgeEvent B Fw Fh ηw ηh}=0 := by |
| apply vague_open_null l _ ν hconv ((isOpen_arrayBridgeEvent B Fw Fh ηw ηh hFw.continuous hFh.continuous).preimage continuous_subtype_val) |
| intro g hg hz |
| exact (S.bridge_vague_tendsto B p len hp hpl Fw Fh L hFw hFh hFwz hFhz hW0 hH0 T hT ηw ηh hηw hηh g hg hz).mono_left hl |
| end CriticalScaleSequence |
| end StandardMapEntropy |
| |
| end OAI |
| |
