| import OAI.Dynamics.StandardMap.EntropyEndpoint |
| import OAI.Dynamics.StandardMap.Entropy.PhysicalVolume |
| import OAI.Dynamics.StandardMap.Holonomy.FinePlaqueParameters |
|
|
| namespace OAI |
|
|
| section |
| section |
| namespace StandardMapEntropy |
| open MeasureTheory Set Filter Topology |
| open scoped Topology NNReal ENNReal |
|
|
| lemma compact_fibre_descent {X : Type*} [TopologicalSpace X] [CompactSpace X] |
| {f g : X → ℝ} (hf : Continuous f) (hg : Continuous g) |
| (hfg : ∀ x y, f x=f y → g x=g y) : |
| ∃ H : ℝ → ℝ, ContinuousOn H (range f) ∧ ∀ x, H (f x)=g x := by |
| classical |
| let H : ℝ → ℝ := fun y => if hy : y∈range f then g (Classical.choose hy) else 0 |
| have hH (x : X) : H (f x)=g x := by |
| dsimp only [H] |
| rw [dite_eq_left (mem_range_self x)] |
| exact hfg _ x (Classical.choose_spec (mem_range_self x)) |
| refine ⟨H,?_,hH⟩ |
| let q : X → range f := fun x => ⟨f x,mem_range_self x⟩ |
| have hq : Continuous q := hf.subtype_mk _ |
| have hqs : Function.Surjective q := by |
| rintro ⟨y,x,rfl⟩ |
| exact ⟨x,rfl⟩ |
| have hquot := IsQuotientMap.of_surjective_continuous hqs hq |
| apply continuousOn_iff_continuous_domRestrict.mpr |
| apply hquot.continuous_iff.mpr |
| have he : (fun y : range f => H y) ∘ q=g := by funext x; exact hH x |
| change Continuous ((fun y : range f => H y) ∘ q) |
| rw [he] |
| exact hg |
|
|
| theorem compact_actual_stable_holonomy {X : Type*} [TopologicalSpace X] [CompactSpace X] |
| (k : ℝ) (hk : 0≤k) (χ ε δ : ℝ) (hε : 0≤ε) (hδ : 0<δ) (hδ' : δ≤1/2) |
| (hq : Real.exp (-χ+ε)+δ<1) (hslow : Real.exp (4*ε)*(Real.exp (-χ+ε)+δ)<1) |
| (P Q W : ℂ) (f g : X → ℝ) (hf : Continuous f) (hg : Continuous g) |
| (hfib : ∀ x y, f x=f y ↔ g x=g y) |
| (hplaque : ∀ x, ∃ (v : ℂ) |
| (hv : ∀ n : ℕ, FineRegular k χ ε (complexProjection ((standardLift k)^[n] v))) (s u : ℝ), |
| |s|≤1/12 ∧ |u|≤1/12 ∧ |
| |(fineInverse k χ ε δ (complexProjection v) W).1|≤ |
| |(fineInverse k χ ε δ (complexProjection v) W).2| ∧ |
| (fineInverse k χ ε δ (complexProjection v) W).2≠0 ∧ |
| P+f x•W=fineStableCurve k χ ε δ hδ hq v hv s ∧ |
| Q+g x•W=fineStableCurve k χ ε δ hδ hq v hv u) : |
| ∃ H J : ℝ → ℝ, |
| ContinuousOn H (range f) ∧ ContinuousOn J (range g) ∧ |
| (∀ x, H (f x)=g x ∧ J (g x)=f x) ∧ |
| (∀ y∈range f, J (H y)=y) ∧ (∀ y∈range g, H (J y)=y) ∧ |
| (∀ Z : Set ℝ, Z⊆range f → volume Z=0 → volume (H '' Z)=0) ∧ |
| (∀ Z : Set ℝ, Z⊆range g → volume Z=0 → volume (J '' Z)=0) := by |
| obtain ⟨H,hH,hHval⟩ := compact_fibre_descent hf hg (fun x y h => (hfib x y).mp h) |
| obtain ⟨J,hJ,hJval⟩ := compact_fibre_descent hg hf (fun x y h => (hfib x y).mpr h) |
| refine ⟨H,J,hH,hJ,fun x => ⟨hHval x,hJval x⟩,?_,?_,?_,?_⟩ |
| · rintro y ⟨x,rfl⟩; rw [hHval,hJval] |
| · rintro y ⟨x,rfl⟩; rw [hJval,hHval] |
| · intro Z hZS hZ |
| apply actual_stable_holonomy_null_image k hk χ ε δ hε hδ hδ' hq hslow P Q W (range f) H hH _ hZS hZ |
| rintro t ⟨x,rfl⟩ |
| obtain ⟨v,hv,s,u,hs,hu,hc,hn,hp,hq'⟩ := hplaque x |
| exact ⟨v,hv,s,u,hs,hu,hc,hn,hp,by rw [hHval]; exact hq'⟩ |
| · intro Z hZS hZ |
| apply actual_stable_holonomy_null_image k hk χ ε δ hε hδ hδ' hq hslow Q P W (range g) J hJ _ hZS hZ |
| rintro t ⟨x,rfl⟩ |
| obtain ⟨v,hv,s,u,hs,hu,hc,hn,hp,hq'⟩ := hplaque x |
| exact ⟨v,hv,u,s,hu,hs,hc,hn,hq',by rw [hJval]; exact hp⟩ |
|
|
| end StandardMapEntropy |
|
|
| end |
| section |
| namespace StandardMapEntropy |
| open MeasureTheory Set Filter Topology |
| open scoped Topology NNReal ENNReal |
| open NonlinearStable |
|
|
| lemma fine_transverse_cone (k χ ε δ : ℝ) (w v : ℂ) |
| (hc : ‖(fineInverse k χ ε δ (complexProjection v)).comp (fineFrame k χ ε δ (complexProjection w))- |
| ContinuousLinearMap.id ℝ Plane‖≤1/4) : |
| let W := fineFrame k χ ε δ (complexProjection w) (0,1) |
| |(fineInverse k χ ε δ (complexProjection v) W).1|≤ |
| |(fineInverse k χ ε δ (complexProjection v) W).2| ∧ |
| (fineInverse k χ ε δ (complexProjection v) W).2≠0 := by |
| dsimp only |
| let A := (fineInverse k χ ε δ (complexProjection v)).comp (fineFrame k χ ε δ (complexProjection w)) |
| have h := (ContinuousLinearMap.le_opNorm (A-ContinuousLinearMap.id ℝ Plane) ((0,1) : Plane)).trans |
| (mul_le_mul_of_nonneg_right hc (norm_nonneg _)) |
| have hnorm : ‖((0,1) : Plane)‖=1 := by simp [Prod.norm_def] |
| rw [hnorm,mul_one] at h |
| have he : ((A-ContinuousLinearMap.id ℝ Plane) (0,1))=A (0,1)-(0,1) := rfl |
| rw [he] at h |
| have h1 : |(A (0,1)).1|≤1/4 := by simpa using (norm_fst_le _).trans h |
| have h2 : |(A (0,1)).2-1|≤1/4 := (norm_snd_le _).trans h |
| have h2' : 3/4≤(A (0,1)).2 := by linarith [(abs_le.mp h2).1] |
| change |(A (0,1)).1|≤|(A (0,1)).2| ∧ (A (0,1)).2≠0 |
| exact ⟨h1.trans (by linarith [le_abs_self ((A (0,1)).2)]),ne_of_gt (by linarith)⟩ |
|
|
| theorem actual_compact_slice_holonomy {X : Type*} [TopologicalSpace X] [CompactSpace X] [T2Space X] |
| (k : ℝ) (hk : 0≤k) (χ ε δ : ℝ) (hε : 0≤ε) (hδ : 0<δ) (hδ' : δ≤1/2) |
| (hq : Real.exp (-χ+ε)+δ<1) (hslow : Real.exp (4*ε)*(Real.exp (-χ+ε)+δ)<1) |
| (w : ℂ) (hw : wedge (stableVector k (complexProjection w)) (unstableVector k (complexProjection w))≠0) |
| (v : X → ℂ) (hvc : Continuous v) |
| (hv : ∀ x, ∀ n : ℕ, FineRegular k χ ε (complexProjection ((standardLift k)^[n] (v x)))) |
| (hdata : ∀ n : ℕ, Continuous (fun x => lyapunovChartData k χ ε |
| (complexProjection ((standardLift k)^[n] (v x))))) |
| {r R : ℝ} (hr : 0<r) (hrsmall : r≤1/12) (hR : 0<R) |
| (hRv : ∀ x, R≤codingRadius k χ ε δ (complexProjection (v x))) |
| (hsize : (4*fineScale k ε δ*r)/(100*R)+r≤1) |
| (hclose : ∀ x, ‖(fineInverse k χ ε δ (complexProjection w)).comp (fineFrame k χ ε δ (complexProjection (v x)))- |
| ContinuousLinearMap.id ℝ Plane‖≤1/4) |
| (hclose' : ∀ x, ‖(fineInverse k χ ε δ (complexProjection (v x))).comp (fineFrame k χ ε δ (complexProjection w))- |
| ContinuousLinearMap.id ℝ Plane‖≤1/4) |
| (hcentre : ∀ x, |(fineInverse k χ ε δ (complexProjection w) (v x-w)).1|≤r/4) : |
| ∃ σ : (X×Icc (-r/2) (r/2)) → Icc (-r) r, Continuous σ ∧ |
| (∀ p : X×Icc (-r/2) (r/2), |
| (finePlaqueInChart k χ ε δ hδ hq w (v p.1) (hv p.1) (σ p)).1=(p.2 : ℝ)) ∧ |
| ∀ c d : Icc (-r/2) (r/2), |
| let f := fun x => (finePlaqueInChart k χ ε δ hδ hq w (v x) (hv x) (σ (x,c))).2 |
| let g := fun x => (finePlaqueInChart k χ ε δ hδ hq w (v x) (hv x) (σ (x,d))).2 |
| ∃ H J : ℝ → ℝ, |
| ContinuousOn H (range f) ∧ ContinuousOn J (range g) ∧ |
| (∀ x, H (f x)=g x ∧ J (g x)=f x) ∧ |
| (∀ y∈range f, J (H y)=y) ∧ (∀ y∈range g, H (J y)=y) ∧ |
| (∀ Z : Set ℝ, Z⊆range f → volume Z=0 → volume (H '' Z)=0) ∧ |
| (∀ Z : Set ℝ, Z⊆range g → volume Z=0 → volume (J '' Z)=0) := by |
| have hr1 : r≤1 := by linarith |
| obtain ⟨σ,hσ,hσval⟩ := fine_compact_graph_crossing k χ ε δ hδ hq w v hvc hv hdata hr hr1 hclose hcentre |
| refine ⟨σ,hσ,hσval,?_⟩ |
| intro c d |
| dsimp only |
| let f (x : X) := (finePlaqueInChart k χ ε δ hδ hq w (v x) (hv x) (σ (x,c))).2 |
| let g (x : X) := (finePlaqueInChart k χ ε δ hδ hq w (v x) (hv x) (σ (x,d))).2 |
| have hs (x : X) (a : Icc (-r/2) (r/2)) : |(σ (x,a) : ℝ)|≤r := abs_le.mpr (σ (x,a)).property |
| have hc (a : Icc (-r/2) (r/2)) : Continuous |
| (fun x => fineStableCurve k χ ε δ hδ hq (v x) (hv x) (σ (x,a))) := by |
| let s : X → Icc (-1 : ℝ) 1 := fun x => ⟨σ (x,a),abs_le.mp ((hs x a).trans hr1)⟩ |
| have hsc : Continuous s := ((hσ.comp (continuous_id.prodMk continuous_const)).subtype_val).subtype_mk _ |
| exact (continuous_fineStableCurve_family k χ ε δ hδ hq v hvc hv hdata).comp |
| (continuous_id.prodMk hsc) |
| have hf : Continuous f := continuous_snd.comp ((fineInverse k χ ε δ (complexProjection w)).continuous.comp |
| ((hc c).sub continuous_const)) |
| have hg : Continuous g := continuous_snd.comp ((fineInverse k χ ε δ (complexProjection w)).continuous.comp |
| ((hc d).sub continuous_const)) |
| have hslow2 : Real.exp (2*ε)*(Real.exp (-χ+ε)+δ)≤1 := by |
| apply le_trans _ hslow.le |
| exact mul_le_mul_of_nonneg_right (Real.exp_le_exp.mpr (by linarith)) (by positivity) |
| have meet (a : Icc (-r/2) (r/2)) (x y : X) |
| (he : (finePlaqueInChart k χ ε δ hδ hq w (v x) (hv x) (σ (x,a))).2= |
| (finePlaqueInChart k χ ε δ hδ hq w (v y) (hv y) (σ (y,a))).2) : |
| fineStableCurve k χ ε δ hδ hq (v x) (hv x) (σ (x,a))= |
| fineStableCurve k χ ε δ hδ hq (v y) (hv y) (σ (y,a)) := by |
| have hi : finePlaqueInChart k χ ε δ hδ hq w (v x) (hv x) (σ (x,a))= |
| finePlaqueInChart k χ ε δ hδ hq w (v y) (hv y) (σ (y,a)) := |
| Prod.ext ((hσval (x,a)).trans (hσval (y,a)).symm) he |
| have hi' := congrArg (fineFrame k χ ε δ (complexProjection w)) hi |
| simp only [finePlaqueInChart,fineFrame_inverse k χ ε hδ _ hw] at hi' |
| exact sub_left_injective hi' |
| have hfib : ∀ x y, f x=f y ↔ g x=g y := by |
| intro x y |
| constructor |
| · intro he |
| have hp := fine_small_crossings_coincide k χ ε δ hδ hq hslow2 w (v x) (v y) (hv x) (hv y) |
| hr hr1 hR (hRv x) hsize (hclose x) (hs x c) (hs x d) (hs y c) (hs y d) |
| (meet c x y he) ((hσval (x,d)).trans (hσval (y,d)).symm) |
| exact congrArg (fun p => (fineInverse k χ ε δ (complexProjection w) (p-w)).2) hp |
| · intro he |
| have hp := fine_small_crossings_coincide k χ ε δ hδ hq hslow2 w (v x) (v y) (hv x) (hv y) |
| hr hr1 hR (hRv x) hsize (hclose x) (hs x d) (hs x c) (hs y d) (hs y c) |
| (meet d x y he) ((hσval (x,c)).trans (hσval (y,c)).symm) |
| exact congrArg (fun p => (fineInverse k χ ε δ (complexProjection w) (p-w)).2) hp |
| let W := fineFrame k χ ε δ (complexProjection w) (0,1) |
| have endpoint (a : Icc (-r/2) (r/2)) (x : X) : |
| w+fineFrame k χ ε δ (complexProjection w) ((a : ℝ),0)+ |
| (finePlaqueInChart k χ ε δ hδ hq w (v x) (hv x) (σ (x,a))).2•W= |
| fineStableCurve k χ ε δ hδ hq (v x) (hv x) (σ (x,a)) := by |
| let p := finePlaqueInChart k χ ε δ hδ hq w (v x) (hv x) (σ (x,a)) |
| have hp : ((a : ℝ),0)+p.2•((0,1) : Plane)=p := by |
| change ((a : ℝ)+p.2*0,0+p.2*1)=p |
| apply Prod.ext |
| · change (a : ℝ)+p.2*0=p.1 |
| simpa only [mul_zero,add_zero] using (hσval (x,a)).symm |
| · change 0+p.2*1=p.2 |
| ring |
| change w+fineFrame k χ ε δ (complexProjection w) ((a : ℝ),0)+p.2•fineFrame k χ ε δ (complexProjection w) (0,1)=_ |
| rw [← map_smul,add_assoc,← map_add,hp] |
| dsimp only [p,finePlaqueInChart] |
| rw [fineFrame_inverse k χ ε hδ _ hw] |
| abel |
| apply compact_actual_stable_holonomy k hk χ ε δ hε hδ hδ' hq hslow |
| (w+fineFrame k χ ε δ (complexProjection w) ((c : ℝ),0)) |
| (w+fineFrame k χ ε δ (complexProjection w) ((d : ℝ),0)) W f g hf hg hfib |
| intro x |
| exact ⟨v x,hv x,σ (x,c),σ (x,d),(hs x c).trans hrsmall,(hs x d).trans hrsmall, |
| (fine_transverse_cone k χ ε δ w (v x) (hclose' x)).1, |
| (fine_transverse_cone k χ ε δ w (v x) (hclose' x)).2,endpoint c x,endpoint d x⟩ |
|
|
| end StandardMapEntropy |
|
|
| end |
| section |
| namespace StandardMapEntropy |
| open MeasureTheory Set Filter Topology |
| open scoped Topology ENNReal |
|
|
| lemma complexRep_norm_le_two (z : Torus) : ‖complexRep z‖≤2 := by |
| have h1 := circleRep_mem z.1 |
| have h2 := circleRep_mem z.2 |
| apply (Complex.norm_le_abs_re_add_abs_im _).trans |
| change |circleRep z.1|+|circleRep z.2|≤2 |
| rw [abs_of_nonneg h1.1,abs_of_nonneg h2.1] |
| linarith [h1.2,h2.2] |
|
|
| theorem compact_lift_fineRegular_block (k : ℝ) (hk : 0≤k) {χ ε : ℝ} (hχ : 0<χ) (hε : 0<ε) |
| (hgap : 0<area (spectralGapRegion k hk χ)) : |
| ∃ C : Set ℂ, IsCompact C ∧ 0<volume C ∧ |
| (∀ v∈C,complexProjection v∈spectralGapRegion k hk χ) ∧ |
| ContinuousOn (fun v => orbitChartData k χ ε (complexProjection v)) C ∧ |
| ∀ v∈C, ∀ n : ℕ, |
| FineRegular k χ ε (complexProjection ((standardLift k)^[n] v)) ∧ |
| FineRegular k χ ε (complexProjection ((standardInverseLift k)^[n] v)) := by |
| obtain ⟨K,hKG,hK,hKpos,hdata,hreg⟩ := compact_fineRegular_block k hk hχ hε hgap |
| let C : Set ℂ := complexProjection ⁻¹' K ∩ Metric.closedBall 0 2 |
| have hC : IsCompact C := (isCompact_closedBall (0 : ℂ) 2).inter_left |
| (hK.isClosed.preimage continuous_complexProjection) |
| have hKC : area K≤volume C := area_le_volume_of_complexRep_subset hK.measurableSet (by |
| intro z hz |
| exact ⟨by simpa only [mem_preimage,complexProjection_complexRep] using hz, |
| by simpa only [Metric.mem_closedBall,dist_zero_right] using complexRep_norm_le_two z⟩) |
| refine ⟨C,hC,hKpos.trans_le hKC,fun v hv => hKG hv.1,?_,?_⟩ |
| · exact hdata.comp continuous_complexProjection.continuousOn (fun _ hv => hv.1) |
| · intro v hv n |
| have h := hreg (complexProjection v) hv.1 n |
| simpa only [complexProjection_iterate,complexProjection_inverse_iterate] using h |
|
|
| end StandardMapEntropy |
|
|
| end |
| section |
| namespace StandardMapEntropy |
| open MeasureTheory Set Filter Topology |
| open scoped Topology ENNReal |
|
|
| lemma positive_compact_neighborhood {C : Set ℂ} (hC : IsCompact C) (hpos : 0<volume C) : |
| ∃ w∈C, ∀ P : ℂ → Prop, (∀ᶠ v in 𝓝[C] w, P v) → |
| ∃ K : Set ℂ, K⊆C ∧ IsCompact K ∧ 0<volume K ∧ ∀ v∈K,P v := by |
| have hpos' : 0<(volume.restrict C) C := by simpa only [Measure.restrict_apply hC.measurableSet,inter_self] using hpos |
| obtain ⟨w,hwC,hwSupp⟩ := Measure.nonempty_inter_support_of_pos hpos' |
| refine ⟨w,hwC,?_⟩ |
| intro P hP |
| obtain ⟨a,ha,hak⟩ := Metric.mem_nhdsWithin_iff.mp hP |
| let K := C∩Metric.closedBall w (a/2) |
| have hb : Metric.ball w (a/2)∈𝓝 w := Metric.ball_mem_nhds _ (by linarith) |
| have hbp : 0<volume (Metric.ball w (a/2)∩C) := by |
| have h := ((Measure.mem_support_iff_forall w).mp hwSupp) _ hb |
| simpa only [Measure.restrict_apply measurableSet_ball] using h |
| refine ⟨K,inter_subset_left,hC.inter_right Metric.isClosed_closedBall,?_,?_⟩ |
| · apply hbp.trans_le (measure_mono _) |
| rintro v ⟨hv,hvC⟩ |
| exact ⟨hvC,Metric.ball_subset_closedBall hv⟩ |
| · intro v hv |
| apply hak |
| exact ⟨(Metric.mem_ball).mpr (lt_of_le_of_lt (Metric.mem_closedBall.mp hv.2) (by linarith)),hv.1⟩ |
|
|
| end StandardMapEntropy |
|
|
| end |
| section |
| namespace StandardMapEntropy |
| open MeasureTheory Set Filter Topology |
| open scoped Topology ENNReal |
| open NonlinearStable |
|
|
| structure SmallPesinBlock (k χ ε δ : ℝ) where |
| carrier : Set ℂ |
| compact : IsCompact carrier |
| positive : 0<volume carrier |
| centre : ℂ |
| centre_regular : wedge (stableVector k (complexProjection centre)) (unstableVector k (complexProjection centre))≠0 |
| r : ℝ |
| r_pos : 0<r |
| r_small : r≤1/12 |
| R : ℝ |
| R_pos : 0<R |
| radius_lower : ∀ v∈carrier,R≤codingRadius k χ ε δ (complexProjection v) |
| size : (4*fineScale k ε δ*r)/(100*R)+r≤1 |
| data : ContinuousOn (fun v => orbitChartData k χ ε (complexProjection v)) carrier |
| regular : ∀ v∈carrier,∀ n : ℕ, FineRegular k χ ε (complexProjection ((standardLift k)^[n] v)) ∧ |
| FineRegular k χ ε (complexProjection ((standardInverseLift k)^[n] v)) |
| frame_close : ∀ v∈carrier, ‖(fineInverse k χ ε δ (complexProjection centre)).comp (fineFrame k χ ε δ (complexProjection v))- |
| ContinuousLinearMap.id ℝ Plane‖≤1/4 |
| coframe_close : ∀ v∈carrier, ‖(fineInverse k χ ε δ (complexProjection v)).comp (fineFrame k χ ε δ (complexProjection centre))- |
| ContinuousLinearMap.id ℝ Plane‖≤1/4 |
| centre_close : ∀ v∈carrier, |(fineInverse k χ ε δ (complexProjection centre) (v-centre)).1|≤r/4 |
|
|
| theorem smallPesinBlock_within (k χ ε δ : ℝ) (hδ : 0<δ) |
| {C : Set ℂ} (hC : IsCompact C) (hCpos : 0<volume C) |
| (hdata : ContinuousOn (fun v => orbitChartData k χ ε (complexProjection v)) C) |
| (hreg : ∀ v∈C,∀ n : ℕ, FineRegular k χ ε (complexProjection ((standardLift k)^[n] v)) ∧ |
| FineRegular k χ ε (complexProjection ((standardInverseLift k)^[n] v))) : |
| ∃ B : SmallPesinBlock k χ ε δ, B.carrier⊆C := by |
| obtain ⟨w,hwC,hw⟩ := positive_compact_neighborhood hC hCpos |
| have hwreg : wedge (stableVector k (complexProjection w)) (unstableVector k (complexProjection w))≠0 := |
| (hreg w hwC 0).1.1 |
| let R := codingRadius k χ ε δ (complexProjection w)/2 |
| have hR : 0<R := half_pos (codingRadius_pos k χ ε hδ _) |
| let a := (4*fineScale k ε δ)/(100*R) |
| have hscale := fineScale_pos k ε hδ |
| have ha : 0≤a := by dsimp only [a]; positivity |
| let r := 1/(24*(1+a)) |
| have hr : 0<r := by dsimp only [r]; positivity |
| have hrsmall : r≤1/12 := by |
| dsimp only [r] |
| apply (div_le_iff₀ (by positivity : 0<24*(1+a))).mpr |
| nlinarith |
| have hsize : (4*fineScale k ε δ*r)/(100*R)+r≤1 := by |
| have he : (4*fineScale k ε δ*r)/(100*R)+r=r*(1+a) := by dsimp only [a]; ring |
| rw [he] |
| dsimp only [r] |
| have : 1+a≠0 := by positivity |
| field_simp |
| norm_num |
| have hdc : Continuous (fun v : C => orbitChartData k χ ε (complexProjection v)) := |
| continuousOn_iff_continuous_domRestrict.mp hdata |
| have hdata0 : Continuous (fun v : C => lyapunovChartData k χ ε (complexProjection v)) := by |
| change Continuous (fun v : C => (orbitChartData k χ ε (complexProjection v) 0).1) |
| exact ((continuous_apply 0).comp hdc).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 _ hdata0) |
| 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 _ hdata0) |
| have hrad : ContinuousOn (fun v => codingRadius k χ ε δ (complexProjection v)) C := |
| continuousOn_iff_continuous_domRestrict.mpr (continuous_const.mul (hdata0.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 |
| intro p |
| exact fineInverse_frame k χ ε hδ _ hwreg p |
| 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‖<1/4 := by |
| have hc : ContinuousWithinAt (fun v : ℂ => ‖(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<1/4) |
| dsimp only |
| rw [hIF,sub_self,norm_zero] |
| exact eventually_lt_nhds (by norm_num) |
| have hclose' : ∀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 : ℂ => ‖(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) |
| dsimp only |
| rw [hIF,sub_self,norm_zero] |
| exact eventually_lt_nhds (by norm_num) |
| have hRlower : ∀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 hcentre : ∀ᶠ v in 𝓝[C] w, |(fineInverse k χ ε δ (complexProjection w) (v-w)).1|<r/4 := by |
| have hc : Continuous (fun v : ℂ => |(fineInverse k χ ε δ (complexProjection w) (v-w)).1|) := |
| (continuous_fst.comp ((fineInverse k χ ε δ (complexProjection w)).continuous.comp |
| (continuous_id.sub continuous_const))).abs |
| have hc' : Tendsto (fun v : ℂ => |(fineInverse k χ ε δ (complexProjection w) (v-w)).1|) (𝓝[C] w) (𝓝 0) := by |
| simpa only [ContinuousWithinAt,sub_self,map_zero,Prod.fst_zero,abs_zero] using hc.continuousWithinAt (s := C) (x := w) |
| exact hc'.eventually (eventually_lt_nhds (show (0 : ℝ)<r/4 by positivity)) |
| obtain ⟨K,hKC,hK,hKpos,hKP⟩ := hw |
| (fun v => R<codingRadius k χ ε δ (complexProjection v) ∧ |
| ‖(fineInverse k χ ε δ (complexProjection w)).comp (fineFrame k χ ε δ (complexProjection v))- |
| ContinuousLinearMap.id ℝ Plane‖<1/4 ∧ |
| ‖(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 w) (v-w)).1|<r/4) |
| (hRlower.and (hclose.and (hclose'.and hcentre))) |
| exact ⟨{ |
| carrier := K, compact := hK, positive := hKpos, centre := w, centre_regular := hwreg, |
| r := r, r_pos := hr, r_small := hrsmall, R := R, R_pos := hR, |
| radius_lower := fun v hv => (hKP v hv).1.le, size := hsize, |
| data := hdata.mono hKC, regular := fun v hv => hreg v (hKC hv), |
| frame_close := fun v hv => (hKP v hv).2.1.le, |
| coframe_close := fun v hv => (hKP v hv).2.2.1.le, |
| centre_close := fun v hv => (hKP v hv).2.2.2.le },hKC⟩ |
|
|
| theorem exists_smallPesinBlock (k : ℝ) (hk : 0≤k) {χ ε δ : ℝ} |
| (hχ : 0<χ) (hε : 0<ε) (hδ : 0<δ) (hgap : 0<area (spectralGapRegion k hk χ)) : |
| Nonempty (SmallPesinBlock k χ ε δ) := by |
| obtain ⟨C,hC,hCpos,_,hdata,hreg⟩ := compact_lift_fineRegular_block k hk hχ hε hgap |
| obtain ⟨B,_⟩ := smallPesinBlock_within k χ ε δ hδ hC hCpos hdata hreg |
| exact ⟨B⟩ |
|
|
| end StandardMapEntropy |
|
|
| end |
| end |
|
|
| end OAI |
|
|
