File size: 3,586 Bytes
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 | import OAI.Dynamics.StandardMap.VectorGrowth
namespace OAI
open MeasureTheory Set
open scoped ENNReal BigOperators
open MeasureTheory Set Filter Metric
open scoped ENNReal Topology Classical
namespace StandardMapEntropy
noncomputable def torusFastBridge (k : ℝ) (z : Torus) (N : ℕ) : Prop :=
growthBase k^((999/1000:ℝ)*(N:ℝ))≤‖torusSegmentTransfer k z 0 N‖
lemma measurableSet_torusFastBridge (k : ℝ) (N : ℕ) : MeasurableSet {z | torusFastBridge k z N} :=
(isClosed_le continuous_const (continuous_torusSegmentTransfer k 0 N).norm).measurableSet
noncomputable def torusGoodMiddles (k : ℝ) (z : Torus) (N : ℕ) : Finset ℕ := by
classical
exact (bridgeMiddle N).filter (fun i => torusIter k (i:ℤ) z∈middlePrefixEvent k (N-i) i)
lemma torus_good_middles_count (k : ℝ) (hk : 0≤k) (N : ℕ) (hN : 100≤N)
(z : Torus) (hz : torusFastBridge k z N) :
(N:ℝ)/5≤(torusGoodMiddles k z N).card := by
classical
obtain ⟨u,hu,huN⟩ := exists_unit_opNorm (torusSegmentTransfer k z 0 N)
have hu0 : u≠0 := by intro he; simp [he] at hu
have hM : 1 < growthBase k := by linarith [growthBase_ge_four k hk]
have hlog := Real.log_pos hM
let zlog := transferVectorLog (growthBase k) (torusSegmentCoefficient k z 0) u
have htotal : (999/1000:ℝ)*(N:ℝ)≤zlog N-zlog 0 := by
have hh := Real.le_log_of_rpow_le (by linarith : 0 < growthBase k) hz
dsimp only [zlog,transferVectorLog]
change _≤Real.log ‖torusSegmentTransfer k z 0 N u‖/Real.log (growthBase k)-Real.log ‖u‖/Real.log (growthBase k)
rw [huN,hu,Real.log_one,zero_div,sub_zero]
exact (le_div_iff₀ hlog).mpr hh
have hcount := middle_log_good_count zlog N hN (fun j _ =>
transferVectorLog_step (growthBase k) hM _ u hu0 j (by
obtain ⟨w,rfl⟩ := liftProjection_surjective z
rw [torusSegmentCoefficient_lift]
exact liftSegmentCoefficient_bound k hk w 0 (j+1))) htotal
apply hcount.trans
apply Nat.cast_le.mpr
apply Finset.card_le_card
intro i hi
obtain ⟨hi,hg⟩ := Finset.mem_filter.mp hi
apply Finset.mem_filter.mpr
refine ⟨hi,?_⟩
exact torus_middle_prefix_of_vector k hk z N i (bridgeMiddle_bounds N i hN hi).2.2.2.2 u hu0 hg
lemma lintegral_count_cover {X ι : Type*} [MeasurableSpace X] (μ : Measure X) (s : Finset ι)
(E : ι → Set X) (hE : ∀ i∈s, MeasurableSet (E i)) (F : Set X) (hF : MeasurableSet F)
(w : X → ℝ≥0∞) (hw : Measurable w) (a : ℝ)
(hcount : ∀ z∈F, a≤(s.filter (fun i => z∈E i)).card) :
ENNReal.ofReal a*(∫⁻ z in F, w z ∂μ)≤∑ i∈s, ∫⁻ z in E i, w z ∂μ := by
classical
have hpoint (z : X) : ENNReal.ofReal a*F.indicator w z≤∑ i∈s, (E i).indicator w z := by
by_cases hz : z∈F
· rw [Set.indicator_of_mem hz]
have he : (∑ i∈s, (E i).indicator w z)=((s.filter (fun i => z∈E i)).card:ℝ≥0∞)*w z := by
simp only [Set.indicator_apply]
rw [← Finset.sum_filter]
simp only [Finset.sum_const,nsmul_eq_mul]
rw [he]
apply mul_le_mul_of_nonneg_right _ bot_le
simpa only [ENNReal.ofReal_natCast] using ENNReal.ofReal_le_ofReal (hcount z hz)
· simp only [Set.indicator_of_notMem hz,mul_zero]
exact bot_le
have hh := lintegral_mono hpoint (μ:=μ)
rw [lintegral_const_mul _ (hw.indicator hF)] at hh
rw [lintegral_indicator hF] at hh
rw [lintegral_finsetSum _ (fun i hi => hw.indicator (hE i hi))] at hh
apply hh.trans_eq
apply Finset.sum_congr rfl
intro i hi
exact lintegral_indicator (hE i hi) _
end StandardMapEntropy
end OAI
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