| import OAI.Dynamics.StandardMap.VectorGrowth |
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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 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 |
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| end OAI |
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