| import OAI.Dynamics.StandardMap.EntropyEndpoint |
| import OAI.Dynamics.StandardMap.Components.BirkhoffAtoms |
|
|
| namespace OAI |
|
|
| section |
| section |
| open MeasureTheory MeasureTheory.Measure Set Filter |
| open scoped ENNReal |
|
|
| namespace BoundedSubadditive |
| variable {X : Type*} [MeasurableSpace X] {μ : Measure X} [IsFiniteMeasure μ] |
|
|
| structure PowerAtomFamily (e : X ≃ᵐ X) (μ : Measure X) where |
| atom : ℕ → Set X |
| measurable : ∀ n,MeasurableSet (atom n) |
| positive : ∀ n,0<μ (atom n) |
| invariant : ∀ n,(e^[n+1]) ⁻¹' (atom n)=atom n |
| ergodic : ∀ n,Ergodic (e^[n+1]) (μ.restrict (atom n)) |
| common : ∃ D : Set X,0<μ D ∧ ∀ n,D⊆atom n |
|
|
| namespace PowerAtomFamily |
| variable {e : X ≃ᵐ X} (F : PowerAtomFamily e μ) (he : MeasurePreserving e μ μ) |
|
|
| lemma refinement {n m : ℕ} (hdiv : n+1 ∣ m+1) : F.atom m ≤ᵐ[μ] F.atom n := by |
| obtain ⟨D,hD,hDA⟩ := F.common |
| have hi : 0<μ (F.atom m∩F.atom n) := hD.trans_le (measure_mono (fun x hx => ⟨hDA m hx,hDA n hx⟩)) |
| apply ergodic_restrict_subset_of_inter (F.measurable m) (F.measurable n) (F.ergodic m).toPreErgodic _ hi |
| obtain ⟨q,hq⟩ := hdiv |
| simpa only [hq,Function.iterate_mul,Function.IsFixedPt] using Function.IsFixedPt.preimage_iterate (F.invariant n) q |
|
|
| noncomputable def cycle (n : ℕ) : CyclicAtom e μ (F.atom 0) (F.atom n) (n+1) := |
| Classical.choice (finite_cycle_of_ergodic_atom e he (F.measurable 0) (F.measurable n) (F.positive n) |
| (F.refinement (by simp)) (by simpa only [zero_add,Function.iterate_one] using F.invariant 0) |
| (by simpa only [zero_add,Function.iterate_one] using F.ergodic 0) (Nat.succ_pos n) (F.invariant n) (F.ergodic n)) |
|
|
| lemma periods_bounded : BddAbove (Set.range (fun n => (F.cycle he n).period)) := by |
| obtain ⟨D,hD,hDA⟩ := F.common |
| have hDr : 0<(μ D).toReal := ENNReal.toReal_pos hD.ne' (measure_ne_top μ D) |
| obtain ⟨B,hB⟩ := exists_nat_gt ((μ (F.atom 0)).toReal/(μ D).toReal) |
| refine ⟨B,?_⟩ |
| rintro p ⟨n,rfl⟩ |
| have hh : ((F.cycle he n).period : ℝ≥0∞)*μ D≤μ (F.atom 0) := by |
| calc |
| _ ≤ ((F.cycle he n).period : ℝ≥0∞)*μ (F.atom n) := mul_le_mul_right (measure_mono (hDA n)) _ |
| _ = _ := (F.cycle he n).mass |
| have hr := ENNReal.toReal_mono (measure_ne_top μ (F.atom 0)) hh |
| simp only [ENNReal.toReal_mul,ENNReal.toReal_natCast] at hr |
| have hd : ((F.cycle he n).period : ℝ)≤(μ (F.atom 0)).toReal/(μ D).toReal := |
| (le_div_iff₀ hDr).mpr hr |
| exact le_of_lt (by exact_mod_cast hd.trans_lt hB) |
|
|
| lemma exists_max_period : ∃ n : ℕ,∀ m,(F.cycle he m).period≤(F.cycle he n).period := by |
| let S := Set.range (fun n => (F.cycle he n).period) |
| have hb : BddAbove S := F.periods_bounded he |
| have hf : S.Finite := Set.finite_iff_bddAbove.mpr hb |
| obtain ⟨n,hn⟩ := (show S.Nonempty from ⟨(F.cycle he 0).period,mem_range_self 0⟩).csSup_mem hf |
| refine ⟨n,?_⟩ |
| intro m |
| change (F.cycle he n).period=sSup S at hn |
| rw [hn] |
| exact le_csSup hb (mem_range_self m) |
|
|
| theorem exists_totally_ergodic_cycle : ∃ n : ℕ,∀ r : ℕ,0 < r → |
| ∀ j : Fin (F.cycle he n).period, |
| Ergodic ((e^[(F.cycle he n).period])^[r]) |
| (μ.restrict ((e.symm^[j.val]) ⁻¹' F.atom n)) := by |
| obtain ⟨n,hn⟩ := F.exists_max_period he |
| refine ⟨n,?_⟩ |
| intro r hr j |
| let C := F.cycle he n |
| let m := (n+1)*r-1 |
| have hm : m+1=(n+1)*r := by |
| have hpos : 0<(n+1)*r := Nat.mul_pos (Nat.succ_pos n) hr |
| exact Nat.sub_add_cancel hpos |
| have hmn : n+1 ∣ m+1 := ⟨r,hm⟩ |
| have hsub := F.refinement hmn |
| have hmasses : μ (F.atom n)≤μ (F.atom m) := by |
| have hmul : (C.period : ℝ≥0∞)*μ (F.atom n)≤(C.period : ℝ≥0∞)*μ (F.atom m) := by |
| calc |
| _ = μ (F.atom 0) := C.mass |
| _ = ((F.cycle he m).period : ℝ≥0∞)*μ (F.atom m) := (F.cycle he m).mass.symm |
| _ ≤ _ := mul_le_mul_left (by exact_mod_cast hn m) _ |
| exact (ENNReal.mul_le_mul_iff_right (by exact_mod_cast C.positive.ne') (by simp)).mp hmul |
| have heq : F.atom m =ᵐ[μ] F.atom n := ae_eq_of_ae_subset_of_measure_ge hsub hmasses |
| (F.measurable m).nullMeasurableSet (measure_ne_top μ (F.atom n)) |
| have herm : Ergodic (e^[m+1]) (μ.restrict (F.atom n)) := ergodic_restrict_congr (F.ergodic m) heq |
| have hbase : Ergodic (e^[C.period]) (μ.restrict (F.atom n)) := by |
| simpa only [Function.iterate_zero,preimage_id_eq,id_eq] using C.ergodic ⟨0,C.positive⟩ |
| have ht : Ergodic ((e^[C.period])^[r]) (μ.restrict (F.atom n)) := by |
| refine ⟨hbase.toMeasurePreserving.iterate r,?_⟩ |
| obtain ⟨q,hq⟩ := C.divides |
| apply PreErgodic.of_iterate q |
| have hnum : m+1=(C.period*r)*q := by |
| calc |
| m+1=(n+1)*r := hm |
| _ = (C.period*q)*r := congrArg (fun a : ℕ => a*r) hq |
| _ = _ := by ac_rfl |
| simpa only [hnum,Function.iterate_mul] using herm.toPreErgodic |
| have ht' : Ergodic (e^[C.period*r]) (μ.restrict (F.atom n)) := by |
| simpa only [Function.iterate_mul] using ht |
| simpa only [Function.iterate_mul] using ergodic_translate e he (F.measurable n) ht' j.val |
|
|
| end PowerAtomFamily |
| end BoundedSubadditive |
|
|
| end |
| section |
| namespace StandardMapEntropy |
| open MeasureTheory Set Filter Topology |
| open scoped Topology ENNReal |
| open NonlinearStable |
|
|
| theorem exists_reversibleRectangleBlock (k : ℝ) (hk : 0≤k) {χ : ℝ} (hχ : 0<χ) |
| (hgap : 0<area (spectralGapRegion k hk χ)) : Nonempty (ReversibleRectangleBlock k χ) := by |
| obtain ⟨ε,δ₀,hε,hδ₀,hδ₀',hq₀,hslow₀,_⟩ := fine_holonomy_parameters hχ (by norm_num : (0 : ℝ)<1) |
| obtain ⟨C,hC,hCpos,hCG,hdata,hreg⟩ := compact_lift_reversible_block k hk hχ hε hgap |
| obtain ⟨w,hwC,hw⟩ := positive_compact_neighborhood hC hCpos |
| have hwreg : wedge (stableVector k (complexProjection w)) (unstableVector k (complexProjection w))≠0 := |
| (hreg w hwC).1 0 |>.1.1 |
| have hwRreg : wedge (stableVector k (complexProjection (tangentReversal w))) |
| (unstableVector k (complexProjection (tangentReversal w)))≠0 := (hreg w hwC).2.1 0 |>.1.1 |
| obtain ⟨α,β,hα,hβ,hswap⟩ := fine_reversal_frame_swap k χ ε (complexProjection w) hwreg |
| (by simpa only [complexProjection_tangentReversal] using hwRreg) |
| (by simpa only [complexProjection_tangentReversal] using (hreg w hwC).2.2.2.1) |
| (by simpa only [complexProjection_tangentReversal] using (hreg w hwC).2.2.2.2) |
| let η := min (1/16) (|α|/(16*|β|)) |
| have hη : 0<η := lt_min (by norm_num) (div_pos (abs_pos.mpr hα) (by positivity)) |
| have hηsmall : η≤1/16 := min_le_left _ _ |
| have hηscale : (|β|/|α|)*((8/3)*η)≤1/2 := by |
| have hh := (le_div_iff₀ (by positivity : 0<16*|β|)).mp |
| (show η≤|α|/(16*|β|) from min_le_right _ _) |
| have he : (|β|/|α|)*((8/3)*η)=(|β| *((8/3)*η))/|α| := by ring |
| rw [he] |
| apply (div_le_iff₀ (abs_pos.mpr hα)).mpr |
| nlinarith [abs_pos.mpr hα] |
| obtain ⟨δ,hδ,hδ',hq,hslow,hslope⟩ := shrink_fine_holonomy_delta hδ₀ hδ₀' hq₀ hslow₀ hη |
| let R := codingRadius k χ ε δ (complexProjection w)/2 |
| let R' := codingRadius k χ ε δ (complexProjection (tangentReversal w))/2 |
| have hR : 0<R := half_pos (codingRadius_pos k χ ε hδ _) |
| have hR' : 0<R' := half_pos (codingRadius_pos k χ ε hδ _) |
| obtain ⟨r,hr,hrsmall,hsize⟩ := small_chart_radius (fineScale k ε δ) R (fineScale_pos k ε hδ).le hR |
| obtain ⟨r',hr',hrsmall',hsize'⟩ := small_chart_radius (fineScale k ε δ) R' (fineScale_pos k ε hδ).le hR' |
| let ρ := min (r/8) (|α| *r'/8) |
| have hρ : 0<ρ := lt_min (by positivity) (by positivity) |
| have hρr : ρ≤r/8 := min_le_left _ _ |
| have hρr' : ρ≤|α| *r'/8 := min_le_right _ _ |
| have hdc : Continuous (fun v : C => (orbitChartData k χ ε (complexProjection v), |
| orbitChartData k χ ε (complexProjection (tangentReversal v)))) := |
| continuousOn_iff_continuous_domRestrict.mp hdata |
| have hd0 : Continuous (fun v : C => lyapunovChartData k χ ε (complexProjection v)) := |
| ((continuous_apply 0).comp hdc.fst).fst |
| have hdR0 : Continuous (fun v : C => lyapunovChartData k χ ε (complexProjection (tangentReversal v))) := |
| ((continuous_apply 0).comp hdc.snd).fst |
| have hF : ContinuousOn (fun v => fineFrame k χ ε δ (complexProjection v)) C := |
| continuousOn_iff_continuous_domRestrict.mpr (continuous_fineFrame_of_data k χ ε δ _ hd0) |
| have hI : ContinuousOn (fun v => fineInverse k χ ε δ (complexProjection v)) C := |
| continuousOn_iff_continuous_domRestrict.mpr (continuous_fineInverse_of_data k χ ε δ hδ _ hd0) |
| have hFR : ContinuousOn (fun v => fineFrame k χ ε δ (complexProjection (tangentReversal v))) C := |
| continuousOn_iff_continuous_domRestrict.mpr (continuous_fineFrame_of_data k χ ε δ _ hdR0) |
| have hIR : ContinuousOn (fun v => fineInverse k χ ε δ (complexProjection (tangentReversal v))) C := |
| continuousOn_iff_continuous_domRestrict.mpr (continuous_fineInverse_of_data k χ ε δ hδ _ hdR0) |
| have hrad : ContinuousOn (fun v => codingRadius k χ ε δ (complexProjection v)) C := |
| continuousOn_iff_continuous_domRestrict.mpr (continuous_const.mul (hd0.snd.snd.pow 2)) |
| have hradR : ContinuousOn (fun v => codingRadius k χ ε δ (complexProjection (tangentReversal v))) C := |
| continuousOn_iff_continuous_domRestrict.mpr (continuous_const.mul (hdR0.snd.snd.pow 2)) |
| have hIF : (fineInverse k χ ε δ (complexProjection w)).comp (fineFrame k χ ε δ (complexProjection w))= |
| ContinuousLinearMap.id ℝ Plane := by |
| apply ContinuousLinearMap.ext |
| exact fineInverse_frame k χ ε hδ _ hwreg |
| have hIFR : (fineInverse k χ ε δ (complexProjection (tangentReversal w))).comp |
| (fineFrame k χ ε δ (complexProjection (tangentReversal w)))=ContinuousLinearMap.id ℝ Plane := by |
| apply ContinuousLinearMap.ext |
| exact fineInverse_frame k χ ε hδ _ hwRreg |
| have hclose : ∀ᶠ v in 𝓝[C] w,‖(fineInverse k χ ε δ (complexProjection w)).comp |
| (fineFrame k χ ε δ (complexProjection v))-ContinuousLinearMap.id ℝ Plane‖<η := by |
| have hc : ContinuousWithinAt (fun v => ‖(fineInverse k χ ε δ (complexProjection w)).comp |
| (fineFrame k χ ε δ (complexProjection v))-ContinuousLinearMap.id ℝ Plane‖) C w := (((continuousOn_const.clm_comp hF).sub continuousOn_const).norm w hwC) |
| apply hc.eventually (p := fun y : ℝ => y<η) |
| simp only [hIF,sub_self,norm_zero] |
| exact eventually_lt_nhds hη |
| have hcloseR : ∀ᶠ v in 𝓝[C] w,‖(fineInverse k χ ε δ (complexProjection (tangentReversal w))).comp |
| (fineFrame k χ ε δ (complexProjection (tangentReversal v)))-ContinuousLinearMap.id ℝ Plane‖<η := by |
| have hc : ContinuousWithinAt (fun v => ‖(fineInverse k χ ε δ (complexProjection (tangentReversal w))).comp |
| (fineFrame k χ ε δ (complexProjection (tangentReversal v)))-ContinuousLinearMap.id ℝ Plane‖) C w := (((continuousOn_const.clm_comp hFR).sub continuousOn_const).norm w hwC) |
| apply hc.eventually (p := fun y : ℝ => y<η) |
| simp only [hIFR,sub_self,norm_zero] |
| exact eventually_lt_nhds hη |
| have hco : ∀ᶠ v in 𝓝[C] w,‖(fineInverse k χ ε δ (complexProjection v)).comp |
| (fineFrame k χ ε δ (complexProjection w))-ContinuousLinearMap.id ℝ Plane‖<1/4 := by |
| have hc : ContinuousWithinAt (fun v => ‖(fineInverse k χ ε δ (complexProjection v)).comp |
| (fineFrame k χ ε δ (complexProjection w))-ContinuousLinearMap.id ℝ Plane‖) C w := (((hI.clm_comp continuousOn_const).sub continuousOn_const).norm w hwC) |
| apply hc.eventually (p := fun y : ℝ => y<1/4) |
| simp only [hIF,sub_self,norm_zero] |
| exact eventually_lt_nhds (by norm_num) |
| have hcoR : ∀ᶠ v in 𝓝[C] w,‖(fineInverse k χ ε δ (complexProjection (tangentReversal v))).comp |
| (fineFrame k χ ε δ (complexProjection (tangentReversal w)))-ContinuousLinearMap.id ℝ Plane‖<1/4 := by |
| have hc : ContinuousWithinAt (fun v => ‖(fineInverse k χ ε δ (complexProjection (tangentReversal v))).comp |
| (fineFrame k χ ε δ (complexProjection (tangentReversal w)))-ContinuousLinearMap.id ℝ Plane‖) C w := (((hIR.clm_comp continuousOn_const).sub continuousOn_const).norm w hwC) |
| apply hc.eventually (p := fun y : ℝ => y<1/4) |
| simp only [hIFR,sub_self,norm_zero] |
| exact eventually_lt_nhds (by norm_num) |
| have hRl : ∀ᶠ v in 𝓝[C] w,R<codingRadius k χ ε δ (complexProjection v) := by |
| apply (hrad w hwC).eventually |
| exact eventually_gt_nhds (by dsimp only [R]; linarith [codingRadius_pos k χ ε hδ (complexProjection w)]) |
| have hRRl : ∀ᶠ v in 𝓝[C] w,R'<codingRadius k χ ε δ (complexProjection (tangentReversal v)) := by |
| apply (hradR w hwC).eventually |
| exact eventually_gt_nhds (by dsimp only [R']; linarith [codingRadius_pos k χ ε hδ (complexProjection (tangentReversal w))]) |
| have hcentral : ∀ᶠ v in 𝓝[C] w,‖fineInverse k χ ε δ (complexProjection w) (v-w)‖<ρ/100 := by |
| have hc : Continuous (fun v : ℂ => ‖fineInverse k χ ε δ (complexProjection w) (v-w)‖) := by fun_prop |
| apply (hc.continuousWithinAt (s := C) (x := w)).eventually (p := fun y : ℝ => y<ρ/100) |
| simp only [sub_self,map_zero,norm_zero] |
| exact eventually_lt_nhds (by positivity) |
| have hcentralR : ∀ᶠ v in 𝓝[C] w,‖fineInverse k χ ε δ (complexProjection (tangentReversal w)) |
| (tangentReversal v-tangentReversal w)‖<r'/4 := by |
| have hc : Continuous (fun v : ℂ => ‖fineInverse k χ ε δ (complexProjection (tangentReversal w)) |
| (tangentReversal v-tangentReversal w)‖) := by fun_prop |
| apply (hc.continuousWithinAt (s := C) (x := w)).eventually (p := fun y : ℝ => y<r'/4) |
| simp only [sub_self,map_zero,norm_zero] |
| exact eventually_lt_nhds (by positivity) |
| obtain ⟨K,hKC,hK,hKpos,hKP⟩ := hw |
| (fun v => R<codingRadius k χ ε δ (complexProjection v) ∧ |
| R'<codingRadius k χ ε δ (complexProjection (tangentReversal v)) ∧ |
| ‖(fineInverse k χ ε δ (complexProjection w)).comp (fineFrame k χ ε δ (complexProjection v))- |
| ContinuousLinearMap.id ℝ Plane‖<η ∧ |
| ‖(fineInverse k χ ε δ (complexProjection (tangentReversal w))).comp |
| (fineFrame k χ ε δ (complexProjection (tangentReversal v)))-ContinuousLinearMap.id ℝ Plane‖<η ∧ |
| ‖(fineInverse k χ ε δ (complexProjection v)).comp (fineFrame k χ ε δ (complexProjection w))- |
| ContinuousLinearMap.id ℝ Plane‖<1/4 ∧ |
| ‖(fineInverse k χ ε δ (complexProjection (tangentReversal v))).comp |
| (fineFrame k χ ε δ (complexProjection (tangentReversal w)))-ContinuousLinearMap.id ℝ Plane‖<1/4 ∧ |
| ‖fineInverse k χ ε δ (complexProjection w) (v-w)‖<ρ/100 ∧ |
| ‖fineInverse k χ ε δ (complexProjection (tangentReversal w)) (tangentReversal v-tangentReversal w)‖<r'/4) |
| (hRl.and (hRRl.and (hclose.and (hcloseR.and (hco.and (hcoR.and (hcentral.and hcentralR))))))) |
| exact ⟨{ |
| ε := ε,δ := δ,ε_pos := hε,δ_pos := hδ,δ_small := hδ',contraction := hq,slow := hslow, |
| carrier := K,compact := hK,positive := hKpos,centre := w,centre_regular := hwreg,reverse_regular := hwRreg, |
| α := α,β := β,α_ne := hα,β_ne := hβ, |
| swap := by simpa only [complexProjection_tangentReversal] using hswap δ hδ.ne', |
| η := η,η_pos := hη,η_small := hηsmall,slope := hslope,reverse_slope := hηscale, |
| r := r,r' := r',R := R,R' := R',ρ := ρ,r_pos := hr,r'_pos := hr',R_pos := hR,R'_pos := hR',ρ_pos := hρ, |
| r_small := hrsmall,r'_small := hrsmall',ρ_small := hρr,ρ_reverse := hρr',size := hsize,reverse_size := hsize', |
| data := hdata.mono hKC, |
| regular := fun v hv n => ((hreg v (hKC hv)).1 n).1, |
| reversed := fun v hv n => ((hreg v (hKC hv)).2.1 n).1, |
| radius := fun v hv => (hKP v hv).1.le, |
| reverse_radius := fun v hv => (hKP v hv).2.1.le, |
| frames := fun v hv => (hKP v hv).2.2.1.le, |
| reverse_frames := fun v hv => (hKP v hv).2.2.2.1.le, |
| coframes := fun v hv => (hKP v hv).2.2.2.2.1.le, |
| reverse_coframes := fun v hv => (hKP v hv).2.2.2.2.2.1.le, |
| central := fun v hv => (hKP v hv).2.2.2.2.2.2.1.le, |
| reverse_central := fun v hv => (hKP v hv).2.2.2.2.2.2.2.le }⟩ |
|
|
| end StandardMapEntropy |
|
|
| end |
| section |
| namespace StandardMapEntropy |
| open MeasureTheory MeasureTheory.Measure Set Filter Topology |
| open scoped Topology ENNReal |
| open BoundedSubadditive |
|
|
| theorem actual_hyperbolic_ergodic_component (k : ℝ) (hk : 0≤k) {χ : ℝ} (hχ : 0<χ) |
| (hgap : 0<area (spectralGapRegion k hk χ)) : |
| ∃ E : Set Torus,MeasurableSet E ∧ 0<area E ∧ |
| standardMap k ⁻¹' E=E ∧ Ergodic (standardMap k) (normalizedArea E) ∧ |
| (∀ᵐ z ∂normalizedArea E,∃ l : ℝ,0<l ∧ LyapunovSpectrumAt k z l) := by |
| obtain ⟨B⟩ := exists_reversibleRectangleBlock k hk hχ hgap |
| obtain ⟨F⟩ := reversible_graph_families B |
| obtain ⟨R⟩ := reversible_graph_rectangle F hk |
| obtain ⟨C⟩ := exists_continuousBirkhoffCode (measurePreserving_standardMap k) (measurePreserving_inverseMap k) |
| (inverseMap_standardMap k) (standardMap_inverseMap k) |
| obtain ⟨c,hc⟩ := actual_rectangle_hopf_atom hk R C |
| let E := {z | C.value z=c} |
| let H := {z | 0<standardLyapunov k hk z} |
| have hE : MeasurableSet E := C.measurableSet_atom c |
| have hH : MeasurableSet H := measurableSet_lt measurable_const (measurable_standardLyapunov k hk) |
| have hpos : 0<area E := hc.trans_le (measure_mono inter_subset_left) |
| have hTH : standardMap k ⁻¹' H=H := by |
| ext z |
| simp only [H,mem_preimage,mem_ofPred_eq,standardLyapunov_invariant] |
| have herg : Ergodic (standardMap k) (normalizedArea E) := C.ergodic_atom (measurePreserving_standardMap k) hpos |
| have hHinE : 0<normalizedArea E H := by |
| unfold normalizedArea |
| rw [Measure.smul_apply,Measure.restrict_apply hH,smul_eq_mul,inter_comm H E] |
| exact ENNReal.mul_pos (ENNReal.inv_ne_zero.mpr (measure_ne_top area E)) hc.ne' |
| have hAE : ∀ᵐ z ∂normalizedArea E,z∈H := by |
| rcases herg.toPreErgodic.ae_mem_or_ae_notMem hH hTH with h|h |
| · exact h |
| · have hz : normalizedArea E H=0 := by |
| simpa only [ae_iff,not_not,Set.ofPred_mem_eq] using h |
| exact (hHinE.ne' hz).elim |
| have hac : normalizedArea E≪area := smul_absolutelyContinuous.trans Measure.absolutelyContinuous_restrict |
| refine ⟨E,hE,hpos,C.invariant_atom c,herg,?_⟩ |
| filter_upwards [hAE,hac.ae_le (ae_standardLyapunov_spectrum k hk)] with z hz hs |
| exact ⟨standardLyapunov k hk z,hz,hs⟩ |
|
|
| lemma exists_positive_spectral_gap (k : ℝ) (hk : 0≤k) |
| (hpos : 0<area {z | 0<standardLyapunov k hk z}) : |
| ∃ χ : ℝ,0<χ ∧ 0<area (spectralGapRegion k hk χ) := by |
| by_contra h |
| push Not at h |
| have hz (n : ℕ) : area (spectralGapRegion k hk (1/((n : ℝ)+1)))=0 := |
| le_antisymm (h _ (by positivity)) (bot_le) |
| have hsub : {z | 0<standardLyapunov k hk z}⊆⋃ n : ℕ,spectralGapRegion k hk (1/((n : ℝ)+1)) := by |
| intro z hzL |
| obtain ⟨n,hn⟩ := exists_nat_gt (1/standardLyapunov k hk z) |
| apply mem_iUnion.mpr |
| refine ⟨n,?_⟩ |
| change 1/((n : ℝ)+1)<standardLyapunov k hk z |
| apply (div_lt_iff₀ (by positivity : (0:ℝ)<(n : ℝ)+1)).mpr |
| have ha := (div_lt_iff₀ hzL).mp hn |
| nlinarith |
| exact hpos.ne' (measure_mono_null hsub (measure_iUnion_null hz)) |
|
|
| theorem eventually_actual_hyperbolic_ergodic_component : |
| ∃ k₀ : ℝ,0<k₀ ∧ ∀ k : ℝ,k₀≤k → |
| ∃ E : Set Torus,MeasurableSet E ∧ 0<area E ∧ |
| standardMap k ⁻¹' E=E ∧ Ergodic (standardMap k) (normalizedArea E) ∧ |
| (∀ᵐ z ∂normalizedArea E,∃ l : ℝ,0<l ∧ LyapunovSpectrumAt k z l) := by |
| obtain ⟨K,hK,hdef⟩ := eventually_meanDeficit_small (1/8 : ℝ) (by norm_num) |
| refine ⟨K,hK,?_⟩ |
| intro k hk |
| have hk0 : 0≤k := hK.le.trans hk |
| obtain ⟨χ,hχ,hgap⟩ := exists_positive_spectral_gap k hk0 (positive_standardLyapunov_of_deficit k hk0 (hdef k hk)) |
| exact actual_hyperbolic_ergodic_component k hk0 hχ hgap |
|
|
| end StandardMapEntropy |
|
|
| end |
| section |
| namespace StandardMapEntropy |
| open MeasureTheory MeasureTheory.Measure Set Filter Topology |
| open scoped Topology ENNReal |
| open BoundedSubadditive |
|
|
| theorem actual_powerAtomFamily (k : ℝ) (hk : 0≤k) {χ : ℝ} (hχ : 0<χ) |
| (hgap : 0<area (spectralGapRegion k hk χ)) : |
| ∃ F : PowerAtomFamily (standardMap_measurableEquiv k) area, |
| 0<area (F.atom 0∩{z | 0<standardLyapunov k hk z}) := by |
| obtain ⟨B⟩ := exists_reversibleRectangleBlock k hk hχ hgap |
| obtain ⟨G⟩ := reversible_graph_families B |
| obtain ⟨R⟩ := reversible_graph_rectangle G hk |
| obtain ⟨C⟩ := exists_powerBirkhoffCodes k |
| obtain ⟨c,hc⟩ := actual_rectangle_all_powers_atom hk R C |
| let D := {z | ∀ m,(C m).value z=c m}∩{z | 0<standardLyapunov k hk z} |
| let A (n : ℕ) : Set Torus := {z | (C n).value z=c n} |
| have hD (n : ℕ) : D⊆A n := fun _ hx => hx.1 n |
| have hpos (n : ℕ) : 0<area (A n) := hc.trans_le (measure_mono (hD n)) |
| let F : PowerAtomFamily (standardMap_measurableEquiv k) area := { |
| atom := A |
| measurable := fun n => (C n).measurableSet_atom (c n) |
| positive := hpos |
| invariant := fun n => (C n).invariant_atom (c n) |
| ergodic := fun n => ergodic_restrict_of_normalized (hpos n) |
| ((C n).ergodic_atom ((measurePreserving_standardMap k).iterate (n+1)) (hpos n)) |
| common := ⟨D,hc,hD⟩ } |
| exact ⟨F,hc.trans_le (measure_mono (fun _ hx => ⟨hx.1 0,hx.2⟩))⟩ |
|
|
| lemma positive_spectrum_on_ergodic_component (k : ℝ) (hk : 0≤k) {E : Set Torus} |
| (hE : MeasurableSet E) (herg : Ergodic (standardMap k) (normalizedArea E)) |
| (hpos : 0<area (E∩{z | 0<standardLyapunov k hk z})) : |
| ∀ᵐ z ∂normalizedArea E,∃ l : ℝ,0<l ∧ LyapunovSpectrumAt k z l := by |
| let H := {z | 0<standardLyapunov k hk z} |
| have hH : MeasurableSet H := measurableSet_lt measurable_const (measurable_standardLyapunov k hk) |
| have hTH : standardMap k ⁻¹' H=H := by |
| ext z |
| simp only [H,mem_preimage,mem_ofPred_eq,standardLyapunov_invariant] |
| have hHinE : 0<normalizedArea E H := by |
| unfold normalizedArea |
| rw [Measure.smul_apply,Measure.restrict_apply' hE,smul_eq_mul,inter_comm H E] |
| exact ENNReal.mul_pos (ENNReal.inv_ne_zero.mpr (measure_ne_top area E)) hpos.ne' |
| have hAE : ∀ᵐ z ∂normalizedArea E,z∈H := by |
| rcases herg.toPreErgodic.ae_mem_or_ae_notMem hH hTH with h|h |
| · exact h |
| · have hz : normalizedArea E H=0 := by |
| simpa only [ae_iff,not_not,Set.ofPred_mem_eq] using h |
| exact (hHinE.ne' hz).elim |
| have hac : normalizedArea E≪area := smul_absolutelyContinuous.trans Measure.absolutelyContinuous_restrict |
| filter_upwards [hAE,hac.ae_le (ae_standardLyapunov_spectrum k hk)] with z hz hs |
| exact ⟨standardLyapunov k hk z,hz,hs⟩ |
|
|
| theorem actual_cyclic_totally_ergodic_component (k : ℝ) (hk : 0≤k) {χ : ℝ} (hχ : 0<χ) |
| (hgap : 0<area (spectralGapRegion k hk χ)) : |
| ∃ E : Set Torus,MeasurableSet E ∧ 0<area E ∧ |
| (standardMap k ⁻¹' E=E) ∧ Ergodic (standardMap k) (normalizedArea E) ∧ |
| (∀ᵐ z ∂normalizedArea E,∃ l : ℝ,0<l ∧ LyapunovSpectrumAt k z l) ∧ |
| ∃ (N : ℕ) (hN : 0<N) (P : Fin N → Set Torus), |
| (∀ j,MeasurableSet (P j)) ∧ ((⋃ j,P j) =ᵐ[area] E) ∧ |
| (∀ i j,i≠j → area (P i∩P j)=0) ∧ |
| (∀ j,(standardMap k '' P j) =ᵐ[area] P ⟨(j.val+1)%N,Nat.mod_lt _ hN⟩) ∧ |
| (∀ j,area (P j)=area E/(N : ℝ≥0∞) ∧ 0<area (P j)) ∧ |
| (∀ j,∀ r : ℕ,0<r → Ergodic (((standardMap k)^[N])^[r]) (normalizedArea (P j))) := by |
| obtain ⟨F,hFH⟩ := actual_powerAtomFamily k hk hχ hgap |
| have he : MeasurePreserving (standardMap_measurableEquiv k) area area := measurePreserving_standardMap k |
| obtain ⟨n,hn⟩ := F.exists_totally_ergodic_cycle he |
| let C := F.cycle he n |
| let P (j : Fin C.period) : Set Torus := ((standardMap_measurableEquiv k).symm^[j.val]) ⁻¹' F.atom n |
| have hP (j : Fin C.period) : MeasurableSet (P j) := |
| (F.measurable n).preimage ((standardMap_measurableEquiv k).symm.measurable.iterate j.val) |
| have hPmass (j : Fin C.period) : area (P j)=area (F.atom n) := |
| ((he.symm (standardMap_measurableEquiv k)).iterate j.val).measure_preimage (F.measurable n).nullMeasurableSet |
| have hEerg : Ergodic (standardMap k) (normalizedArea (F.atom 0)) := by |
| have hh := (F.ergodic 0).smul_measure (area (F.atom 0))⁻¹ |
| simp only [zero_add,Function.iterate_one] at hh |
| exact hh |
| refine ⟨F.atom 0,F.measurable 0,F.positive 0,?_,hEerg, |
| positive_spectrum_on_ergodic_component k hk (F.measurable 0) hEerg hFH, |
| C.period,C.positive,P,hP,C.cover,C.disjoint,C.forward,?_,?_⟩ |
| · have hh := F.invariant 0 |
| simp only [zero_add,Function.iterate_one] at hh |
| exact hh |
| · intro j |
| rw [hPmass] |
| exact ⟨(ENNReal.eq_div_iff (by exact_mod_cast C.positive.ne') (by simp)).mpr C.mass,F.positive n⟩ |
| · intro j r hr |
| exact (hn r hr j).smul_measure (area (P j))⁻¹ |
|
|
| theorem eventually_actual_cyclic_totally_ergodic_component : |
| ∃ k₀ : ℝ,0<k₀ ∧ ∀ k : ℝ,k₀≤k → |
| ∃ E : Set Torus,MeasurableSet E ∧ 0<area E ∧ |
| (standardMap k ⁻¹' E=E) ∧ Ergodic (standardMap k) (normalizedArea E) ∧ |
| (∀ᵐ z ∂normalizedArea E,∃ l : ℝ,0<l ∧ LyapunovSpectrumAt k z l) ∧ |
| ∃ (N : ℕ) (hN : 0<N) (P : Fin N → Set Torus), |
| (∀ j,MeasurableSet (P j)) ∧ ((⋃ j,P j) =ᵐ[area] E) ∧ |
| (∀ i j,i≠j → area (P i∩P j)=0) ∧ |
| (∀ j,(standardMap k '' P j) =ᵐ[area] P ⟨(j.val+1)%N,Nat.mod_lt _ hN⟩) ∧ |
| (∀ j,area (P j)=area E/(N : ℝ≥0∞) ∧ 0<area (P j)) ∧ |
| (∀ j,∀ r : ℕ,0<r → Ergodic (((standardMap k)^[N])^[r]) (normalizedArea (P j))) := by |
| obtain ⟨K,hK,hdef⟩ := eventually_meanDeficit_small (1/8 : ℝ) (by norm_num) |
| refine ⟨K,hK,?_⟩ |
| intro k hk |
| have hk0 : 0≤k := hK.le.trans hk |
| obtain ⟨χ,hχ,hgap⟩ := exists_positive_spectral_gap k hk0 (positive_standardLyapunov_of_deficit k hk0 (hdef k hk)) |
| exact actual_cyclic_totally_ergodic_component k hk0 hχ hgap |
|
|
| end StandardMapEntropy |
|
|
| end |
| end |
|
|
| end OAI |
|
|
