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92416a2 | 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 | import OAI.Dynamics.StandardMap.BridgeCritical
namespace OAI
open MeasureTheory Set
open scoped ENNReal BigOperators
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
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