| import OAI.Dynamics.StandardMap.EntropyEndpoint |
| import OAI.Dynamics.StandardMap.Holonomy.ConeIntervalDistortion |
|
|
| namespace OAI |
|
|
| section |
| section |
| namespace StandardMapEntropy |
| open MeasureTheory Set Filter |
| open scoped Topology NNReal ENNReal |
| open NonlinearStable |
|
|
| lemma fine_plaque_parameters {χ τ : ℝ} (hχ : 0<χ) (hτ : 0<τ) : |
| ∃ ε δ : ℝ, 0<ε ∧ 0<δ ∧ δ≤1/2 ∧ Real.exp (-χ+ε)+δ<1 ∧ |
| Real.exp (2*ε)*(Real.exp (-χ+ε)+δ)<1 ∧ |
| Real.exp (-χ+ε)*δ/(1-Real.exp (-χ+ε))≤τ := by |
| let ε := χ/10 |
| let ℓ := Real.exp (-χ+ε) |
| let ρ := Real.exp (-χ/2) |
| have hε : 0<ε := by dsimp [ε]; positivity |
| have hℓ0 : 0<ℓ := Real.exp_pos _ |
| have hℓρ : ℓ<ρ := Real.exp_lt_exp.mpr (by dsimp [ε]; linarith) |
| have hρ1 : ρ<1 := Real.exp_lt_one_iff.mpr (by linarith) |
| have hℓ1 : ℓ<1 := hℓρ.trans hρ1 |
| let δ := min ((ρ-ℓ)/2) (min (τ*(1-ℓ)/2) (1/2)) |
| have hδ : 0<δ := lt_min (by linarith) (lt_min (by positivity) (by norm_num)) |
| have hdρ : δ≤(ρ-ℓ)/2 := min_le_left _ _ |
| have hdτ : δ≤τ*(1-ℓ)/2 := (min_le_right _ _).trans (min_le_left _ _) |
| have hd1 : δ≤1/2 := (min_le_right _ _).trans (min_le_right _ _) |
| have hqρ : ℓ+δ≤ρ := by linarith |
| have hslow : Real.exp (2*ε)*ρ<1 := by |
| rw [←Real.exp_add] |
| exact Real.exp_lt_one_iff.mpr (by dsimp [ε]; linarith) |
| refine ⟨ε,δ,hε,hδ,hd1,hqρ.trans_lt hρ1,?_,?_⟩ |
| · exact (mul_le_mul_of_nonneg_left hqρ (Real.exp_pos _).le).trans_lt hslow |
| · apply (div_le_iff₀ (sub_pos.mpr hℓ1)).mpr |
| have hb := mul_le_mul_of_nonneg_right hℓ1.le hδ.le |
| nlinarith |
|
|
| lemma fineStableGraph_slope (k χ ε δ : ℝ) (hδ : 0<δ) (hq : Real.exp (-χ+ε)+δ<1) |
| (w : ℂ) (hw : ∀ n : ℕ, FineRegular k χ ε (complexProjection ((standardLift k)^[n] w))) |
| (s t : ℝ) : |
| |fineStableGraph k χ ε δ hδ hq w hw s-fineStableGraph k χ ε δ hδ hq w hw t|≤ |
| (Real.exp (-χ+ε)*δ/(1-Real.exp (-χ+ε)))*|s-t| := |
| graph_slope_bound (fineLocalRecurrence k χ ε δ hδ w hw).extend hq s t |
|
|
| end StandardMapEntropy |
|
|
| end |
| section |
| namespace StandardMapEntropy |
| open MeasureTheory Set Filter |
| open scoped Topology NNReal ENNReal |
| open NonlinearStable |
|
|
| lemma fine_holonomy_parameters {χ τ : ℝ} (hχ : 0<χ) (hτ : 0<τ) : |
| ∃ ε δ : ℝ, 0<ε ∧ 0<δ ∧ δ≤1/2 ∧ Real.exp (-χ+ε)+δ<1 ∧ |
| Real.exp (4*ε)*(Real.exp (-χ+ε)+δ)<1 ∧ |
| Real.exp (-χ+ε)*δ/(1-Real.exp (-χ+ε))≤τ := by |
| let ε := χ/10 |
| let ℓ := Real.exp (-χ+ε) |
| let ρ := Real.exp (-χ/2) |
| have hε : 0<ε := by dsimp [ε]; positivity |
| have hℓ0 : 0<ℓ := Real.exp_pos _ |
| have hℓρ : ℓ<ρ := Real.exp_lt_exp.mpr (by dsimp [ε]; linarith) |
| have hρ1 : ρ<1 := Real.exp_lt_one_iff.mpr (by linarith) |
| have hℓ1 : ℓ<1 := hℓρ.trans hρ1 |
| let δ := min ((ρ-ℓ)/2) (min (τ*(1-ℓ)/2) (1/2)) |
| have hδ : 0<δ := lt_min (by linarith) (lt_min (by positivity) (by norm_num)) |
| have hdρ : δ≤(ρ-ℓ)/2 := min_le_left _ _ |
| have hdτ : δ≤τ*(1-ℓ)/2 := (min_le_right _ _).trans (min_le_left _ _) |
| have hd1 : δ≤1/2 := (min_le_right _ _).trans (min_le_right _ _) |
| have hqρ : ℓ+δ≤ρ := by linarith |
| have hslow : Real.exp (4*ε)*ρ<1 := by |
| rw [←Real.exp_add] |
| exact Real.exp_lt_one_iff.mpr (by dsimp [ε]; linarith) |
| refine ⟨ε,δ,hε,hδ,hd1,hqρ.trans_lt hρ1,?_,?_⟩ |
| · exact (mul_le_mul_of_nonneg_left hqρ (Real.exp_pos _).le).trans_lt hslow |
| · apply (div_le_iff₀ (sub_pos.mpr hℓ1)).mpr |
| have hb := mul_le_mul_of_nonneg_right hℓ1.le hδ.le |
| nlinarith |
|
|
| lemma NonlinearStable.Recurrence.orbit_solution {ℓ δ : ℝ≥0} (r : Recurrence ℓ δ) |
| (h : ℓ+δ<1) (s : ℝ) (n : ℕ) : r.orbit (r.solution h s 0) n=r.solution h s n := by |
| induction n with |
| | zero => rfl |
| | succ n ih => rw [Recurrence.orbit,ih,solution_orbit] |
|
|
| lemma fine_central_orbit_decay (k χ ε δ : ℝ) (hδ : 0<δ) (hq : Real.exp (-χ+ε)+δ<1) |
| (w : ℂ) (hw : ∀ n : ℕ, FineRegular k χ ε (complexProjection ((standardLift k)^[n] w))) |
| {v : Plane} {s : ℝ} (hs : |s|≤1/12) |
| (he : w+fineFrame k χ ε δ (complexProjection w) v=fineStableCurve k χ ε δ hδ hq w hw s) |
| (n : ℕ) : ‖(fineLocalRecurrence k χ ε δ hδ w hw).extend.orbit v n‖≤ |
| (Real.exp (-χ+ε)+δ)^n/12 := by |
| let r := fineLocalRecurrence k χ ε δ hδ w hw |
| have hs' : |s|≤1 := hs.trans (by norm_num) |
| have hv : v=r.extend.solution hq s 0 := by |
| have hh := congrArg (fun p => fineCoordinate k χ ε δ w p 0) he |
| rw [fineCoordinate_initial k χ ε hδ w (hw 0),fineStableCoordinate k χ ε δ hδ hq w hw hs'] at hh |
| exact hh |
| rw [hv,r.extend.orbit_solution hq] |
| have hdec : ‖r.extend.solution hq s n‖≤(Real.exp (-χ+ε)+δ)^n*|s| := by |
| have hh := solution_exponential_contraction r.extend hq s 0 n |
| rw [solution_zero r.extend hq] at hh |
| change ‖r.extend.solution hq s n-0‖≤(Real.exp (-χ+ε)+δ)^n*|s-0| at hh |
| simpa only [sub_zero] using hh |
| exact hdec.trans ((mul_le_mul_of_nonneg_left hs (by positivity)).trans_eq (by ring)) |
|
|
| theorem fine_stable_matching_local_bound (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 : ∀ n : ℕ, FineRegular k χ ε (complexProjection ((standardLift k)^[n] w))) |
| (p q V : Plane) (hc : |V.1|≤|V.2|) (hV : V.2≠0) |
| (S : Set ℝ) (H : ℝ → ℝ) {t₀ : ℝ} (ht₀ : t₀∈S) (hH : ContinuousWithinAt H S t₀) |
| {s₀ s₁ : ℝ} (hs₀ : |s₀|≤1/12) (hs₁ : |s₁|≤1/12) |
| (hp₀ : w+fineFrame k χ ε δ (complexProjection w) (p+t₀•V)=fineStableCurve k χ ε δ hδ hq w hw s₀) |
| (hq₀ : w+fineFrame k χ ε δ (complexProjection w) (q+H t₀•V)=fineStableCurve k χ ε δ hδ hq w hw s₁) |
| (hmatch : ∀ t∈S, ∃ (v : ℂ) |
| (hv : ∀ n : ℕ, FineRegular k χ ε (complexProjection ((standardLift k)^[n] v))) (s u : ℝ), |
| |s|≤1 ∧ |u|≤1 ∧ |
| w+fineFrame k χ ε δ (complexProjection w) (p+t•V)=fineStableCurve k χ ε δ hδ hq v hv s ∧ |
| w+fineFrame k χ ε δ (complexProjection w) (q+H t•V)=fineStableCurve k χ ε δ hδ hq v hv u) : |
| ∀ᶠ t in 𝓝[S] t₀, |H t-H t₀|≤(2*Real.exp (8*δ/(1-(Real.exp (-χ+ε)+δ))))* |t-t₀| := by |
| let r := fineLocalRecurrence k χ ε δ hδ w hw |
| let D (n : ℕ) := fineRemainderDerivative k χ ε δ ((standardLift k)^[n] w) |
| let a := Real.exp (2*ε) |
| let b := Real.exp (-χ+ε)+δ |
| have hb : 0<b := add_pos (Real.exp_pos _) hδ |
| have hscale := fineScale_pos k ε hδ |
| have hab : a*b<1 := (mul_le_mul_of_nonneg_right |
| (Real.exp_le_exp.mpr (by linarith : 2*ε≤4*ε)) hb.le).trans_lt hslow |
| have haρ : a*(a*b)<1 := by |
| have he : a*(a*b)=Real.exp (4*ε)*b := by |
| dsimp only [a] |
| rw [←mul_assoc,←Real.exp_add] |
| congr 2 |
| ring |
| rwa [he] |
| obtain ⟨M,hM,hstep⟩ := fine_extended_step_bound k hk χ ε δ hε hδ w hw |
| have hsmall (n : ℕ) (v : Plane) (_ : ‖v‖≤1) : ‖D n v‖≤δ*‖v‖ := by |
| have hh := fineRemainderDerivative_lipschitz k χ ε hδ ((standardLift k)^[n] w) |
| (hw n) (fine_regular_lift_next k χ ε w hw n) v 0 |
| simpa only [fineRemainderDerivative_zero,sub_zero] using hh |
| have htrack (t : ℝ) (ht : t∈S) (N : ℕ) |
| (hP : ∀ n : ℕ,n≤N → ‖r.extend.orbit (p+t•V) n‖≤1/3) |
| (hQ : ∀ n : ℕ,n≤N → ‖r.extend.orbit (q+H t•V) n‖≤1/3) : |
| ‖r.extend.orbit (p+t•V) N-r.extend.orbit (q+H t•V) N‖≤ |
| ((4*fineScale k ε δ)/(100*codingRadius k χ ε δ (complexProjection w)))*(a*b)^N := by |
| have htp : p+t•V∈r.extend.trappedBox N := fun n hn => (hP n hn).trans (by norm_num) |
| have htq : q+H t•V∈r.extend.trappedBox N := fun n hn => (hQ n hn).trans (by norm_num) |
| rw [←r.trueOrbit_eq_extend htp le_rfl,←r.trueOrbit_eq_extend htq le_rfl] |
| rw [fine_trueOrbit,fine_trueOrbit] |
| obtain ⟨v,hv,s,u,hs,hu,hps,hqu⟩ := hmatch t ht |
| have hd (n : ℕ) : |
| ‖(standardLift k)^[n] (w+fineFrame k χ ε δ (complexProjection w) (p+t•V))- |
| (standardLift k)^[n] (w+fineFrame k χ ε δ (complexProjection w) (q+H t•V))‖≤ |
| (4*fineScale k ε δ)*b^n := by |
| rw [hps,hqu] |
| have hh := fineStableCurve_contraction k χ ε δ hδ hq v hv hs hu n |
| have hsu : |s-u|≤2 := (abs_sub _ _).trans (by linarith) |
| have hm := mul_le_mul_of_nonneg_left hsu (by positivity : 0≤(2*fineScale k ε δ)*b^n) |
| exact hh.trans (hm.trans_eq (by ring)) |
| exact fineCoordinate_tracking_rate k χ ε δ hδ w _ _ hw |
| (by positivity [fineScale_pos k ε hδ]) hb.le hd N |
| exact r.holonomy_local_bound hq hδ' D (fun n v => hasFDerivAt_fineRemainder k χ ε δ _ v) |
| hsmall p q V hc hV S H ht₀ hH hM (Real.exp_pos _) hb.le hq.le |
| (mul_nonneg (Real.exp_pos _).le hb.le) hab haρ hstep |
| (fine_central_orbit_decay k χ ε δ hδ hq w hw hs₀ hp₀) |
| (fine_central_orbit_decay k χ ε δ hδ hq w hw hs₁ hq₀) htrack |
|
|
| end StandardMapEntropy |
|
|
| end |
| section |
| namespace StandardMapEntropy |
| open MeasureTheory Set Filter |
| open scoped Topology NNReal ENNReal |
|
|
| theorem null_image_of_pointwise_local_bound {S : Set ℝ} {H : ℝ → ℝ} {K : ℝ} (hK : 0≤K) |
| (h : ∀ x∈S, ∀ᶠ y in 𝓝[S] x, |H y-H x| ≤ K*|y-x|) |
| {Z : Set ℝ} (hZS : Z⊆S) (hZ : volume Z=0) : volume (H '' Z)=0 := by |
| classical |
| let A (n : ℕ) : Set ℝ := {x | x∈S ∧ |H x| ≤ (n : ℝ) ∧ |
| ∀ y∈S, |y-x| < 1/((n : ℝ)+1) → |H y-H x| ≤ K*|y-x|} |
| have hcover : S⊆⋃ n : ℕ,A n := by |
| intro x hx |
| obtain ⟨ε,hε,hball⟩ := Metric.mem_nhdsWithin_iff.mp (h x hx) |
| obtain ⟨n,hn⟩ := exists_nat_gt (max |H x| (1/ε)) |
| have hbn : |H x| ≤ (n : ℝ) := ((le_max_left _ _).trans_lt hn).le |
| have hεn : 1/((n : ℝ)+1)<ε := by |
| have he : 1/ε<(n : ℝ) := (le_max_right _ _).trans_lt hn |
| have hh : 1<(n : ℝ)*ε := (div_lt_iff₀ hε).mp he |
| apply (div_lt_iff₀ (by positivity : 0<(n : ℝ)+1)).mpr |
| nlinarith |
| refine mem_iUnion.mpr ⟨n,hx,hbn,fun y hy hxy => ?_⟩ |
| exact hball ⟨by simpa only [Metric.mem_ball,Real.dist_eq] using hxy.trans hεn,hy⟩ |
| have hLip (n : ℕ) : LipschitzOnWith |
| ⟨max K (2*(n : ℝ)*((n : ℝ)+1)), hK.trans (le_max_left _ _)⟩ H (A n) := by |
| refine LipschitzOnWith.of_dist_le_mul fun x hx y hy => ?_ |
| change |H x-H y| ≤ max K (2*(n : ℝ)*((n : ℝ)+1))*|x-y| |
| by_cases hxy : |x-y| < 1/((n : ℝ)+1) |
| · exact (hy.2.2 x hx.1 hxy).trans (mul_le_mul_of_nonneg_right (le_max_left _ _) (abs_nonneg _)) |
| · have hd : 1/((n : ℝ)+1)≤|x-y| := le_of_not_gt hxy |
| have hprod : 1≤|x-y| *((n : ℝ)+1) := (div_le_iff₀ (by positivity)).mp hd |
| have hleft : |H x-H y| ≤ 2*(n : ℝ) := (abs_sub _ _).trans (by linarith [hx.2.1,hy.2.1]) |
| have hm := mul_le_mul_of_nonneg_left hprod (by positivity : 0≤2*(n : ℝ)) |
| apply hleft.trans |
| apply le_trans _ (mul_le_mul_of_nonneg_right (le_max_right _ _) (abs_nonneg _)) |
| nlinarith |
| have hpieces (n : ℕ) : volume (H '' (Z∩A n))=0 := by |
| have hz : volume (Z∩A n)=0 := measure_mono_null inter_subset_left hZ |
| have hl := ((hLip n).mono (show Z∩A n⊆A n from inter_subset_right)).hausdorffMeasure_image_le (by norm_num : (0:ℝ)≤1) |
| rw [hausdorffMeasure_real,hz,mul_zero] at hl |
| exact bot_unique hl |
| have hsub : H '' Z⊆⋃ n : ℕ,H '' (Z∩A n) := by |
| rintro _ ⟨x,hx,rfl⟩ |
| obtain ⟨n,hn⟩ := mem_iUnion.mp (hcover (hZS hx)) |
| exact mem_iUnion.mpr ⟨n,x,⟨hx,hn⟩,rfl⟩ |
| exact measure_mono_null hsub (measure_iUnion_null hpieces) |
|
|
| end StandardMapEntropy |
|
|
| end |
| section |
| namespace StandardMapEntropy |
| open MeasureTheory Set Filter |
| open scoped Topology NNReal ENNReal |
| open NonlinearStable |
|
|
| lemma fine_affine_coordinates (k χ ε δ : ℝ) (hδ : 0<δ) (w p V : ℂ) |
| (hw : FineRegular k χ ε (complexProjection w)) (t : ℝ) : |
| w+fineFrame k χ ε δ (complexProjection w) |
| (fineInverse k χ ε δ (complexProjection w) (p-w)+t•fineInverse k χ ε δ (complexProjection w) V)= |
| p+t•V := by |
| rw [map_add,map_smul,fineFrame_inverse k χ ε hδ _ hw.1, |
| fineFrame_inverse k χ ε hδ _ hw.1] |
| abel |
|
|
| theorem actual_stable_holonomy_null_image (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) (S : Set ℝ) (H : ℝ → ℝ) (hH : ContinuousOn H S) |
| (hmatch : ∀ t∈S, ∃ (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+t•W=fineStableCurve k χ ε δ hδ hq v hv s ∧ |
| Q+H t•W=fineStableCurve k χ ε δ hδ hq v hv u) |
| {Z : Set ℝ} (hZS : Z⊆S) (hZ : volume Z=0) : volume (H '' Z)=0 := by |
| apply null_image_of_pointwise_local_bound (K := 2*Real.exp (8*δ/(1-(Real.exp (-χ+ε)+δ)))) (by positivity) _ hZS hZ |
| intro t ht |
| obtain ⟨v,hv,s,u,hs,hu,hcone,hnonzero,hleft,hright⟩ := hmatch t ht |
| let I := fineInverse k χ ε δ (complexProjection v) |
| have he (p : ℂ) (t : ℝ) : v+fineFrame k χ ε δ (complexProjection v) (I (p-v)+t•I W)=p+t•W := |
| fine_affine_coordinates k χ ε δ hδ v p W (hv 0) t |
| have hleft' : v+fineFrame k χ ε δ (complexProjection v) (I (P-v)+t•I W)= |
| fineStableCurve k χ ε δ hδ hq v hv s := (he P t).trans hleft |
| have hright' : v+fineFrame k χ ε δ (complexProjection v) (I (Q-v)+H t•I W)= |
| fineStableCurve k χ ε δ hδ hq v hv u := (he Q (H t)).trans hright |
| apply fine_stable_matching_local_bound k hk χ ε δ hε hδ hδ' hq hslow v hv |
| (I (P-v)) (I (Q-v)) (I W) hcone hnonzero S H ht (hH t ht) hs hu hleft' hright' |
| intro t' ht' |
| obtain ⟨v',hv',s',u',hs',hu',_,_,hleft'',hright''⟩ := hmatch t' ht' |
| exact ⟨v',hv',s',u',hs'.trans (by norm_num),hu'.trans (by norm_num), |
| (he P t').trans hleft'',(he Q (H t')).trans hright''⟩ |
|
|
| end StandardMapEntropy |
|
|
| end |
| section |
| namespace StandardMapEntropy |
| open MeasureTheory Set Filter |
| open scoped Topology NNReal ENNReal |
| open NonlinearStable |
|
|
| lemma strictMono_of_linear_error {f : ℝ → ℝ} {κ : ℝ} (hκ : κ<1) |
| (h : ∀ s t : ℝ, |f s-f t-(s-t)|≤κ*|s-t|) : StrictMono f := by |
| intro s t hst |
| have hh := (abs_le.mp (h t s)).1 |
| rw [abs_of_pos (sub_pos.mpr hst)] at hh |
| nlinarith |
|
|
| theorem compact_graph_crossing {X : Type*} [TopologicalSpace X] [CompactSpace X] [T2Space X] |
| (r : ℝ) (hr : 0<r) (F : X → ℝ → Plane) |
| (hF : Continuous (fun p : X×Icc (-r) r => F p.1 p.2)) |
| (he : ∀ x s t, |(F x s).1-(F x t).1-(s-t)|≤(1/4)*|s-t|) |
| (hzero : ∀ x, |(F x 0).1|≤r/4) : |
| ∃ σ : (X×Icc (-r/2) (r/2)) → Icc (-r) r, Continuous σ ∧ |
| ∀ v, (F v.1 (σ v)).1=(v.2 : ℝ) := by |
| classical |
| have hm (x : X) : StrictMono (fun s => (F x s).1) := strictMono_of_linear_error (by norm_num) (he x) |
| have hf (x : X) : ContinuousOn (fun s => (F x s).1) (Icc (-r) r) := by |
| apply continuousOn_iff_continuous_domRestrict.mpr |
| exact (hF.comp (continuous_const.prodMk continuous_id)).fst |
| have hend (x : X) : (F x (-r)).1≤-r/2 ∧ r/2≤(F x r).1 := by |
| have hl := (abs_le.mp (he x (-r) 0)).2 |
| have hu := (abs_le.mp (he x r 0)).1 |
| have hz := abs_le.mp (hzero x) |
| simp only [sub_zero,abs_neg,abs_of_pos hr] at hl hu |
| constructor <;> linarith |
| have hex (v : X×Icc (-r/2) (r/2)) : ∃ s : Icc (-r) r, (F v.1 s).1=(v.2 : ℝ) := by |
| obtain ⟨s,hs,hs'⟩ := intermediate_value_Icc (by linarith : -r≤r) (hf v.1) |
| ⟨(hend v.1).1.trans v.2.property.1,v.2.property.2.trans (hend v.1).2⟩ |
| exact ⟨⟨s,hs⟩,hs'⟩ |
| choose σ hσ using hex |
| let G : X×Icc (-r) r → X×ℝ := fun p => (p.1,(F p.1 p.2).1) |
| have hGc : Continuous G := continuous_fst.prodMk hF.fst |
| have hGi : Function.Injective G := by |
| rintro ⟨x,s⟩ ⟨y,t⟩ heq |
| have hxy : x=y := congrArg Prod.fst heq |
| subst y |
| have hst : (s : ℝ)=(t : ℝ) := (hm x).injective (congrArg Prod.snd heq) |
| exact Prod.ext rfl (Subtype.ext hst) |
| have hemb := hGc.isClosedEmbedding hGi |>.isEmbedding |
| have hpair : Continuous (fun v : X×Icc (-r/2) (r/2) => (v.1,σ v)) := by |
| apply hemb.continuous_iff.mpr |
| have heq : G ∘ (fun v : X×Icc (-r/2) (r/2) => (v.1,σ v))= |
| (fun v => (v.1,(v.2 : ℝ))) := by |
| funext v; exact Prod.ext rfl (hσ v) |
| rw [heq] |
| exact continuous_fst.prodMk (continuous_subtype_val.comp continuous_snd) |
| exact ⟨σ,hpair.snd,hσ⟩ |
|
|
| end StandardMapEntropy |
|
|
| end |
| section |
| namespace StandardMapEntropy |
| open MeasureTheory Set Filter |
| open scoped Topology NNReal ENNReal |
| open NonlinearStable |
|
|
| lemma fine_graph_pair_bound (k χ ε δ : ℝ) (hδ : 0<δ) (hq : Real.exp (-χ+ε)+δ<1) |
| (v : ℂ) (hv : ∀ n : ℕ, FineRegular k χ ε (complexProjection ((standardLift k)^[n] v))) (s t : ℝ) : |
| ‖((s,fineStableGraph k χ ε δ hδ hq v hv s) : Plane)- |
| (t,fineStableGraph k χ ε δ hδ hq v hv t)‖≤|s-t| := by |
| have hg := (graph_lipschitz (fineLocalRecurrence k χ ε δ hδ v hv).extend hq).dist_le_mul s t |
| change max |s-t| |fineStableGraph k χ ε δ hδ hq v hv s-fineStableGraph k χ ε δ hδ hq v hv t|≤|s-t| |
| exact max_le le_rfl (by simpa only [Real.dist_eq,NNReal.coe_one,one_mul,fineStableGraph] using hg) |
|
|
| noncomputable def finePlaqueInChart (k χ ε δ : ℝ) (hδ : 0<δ) (hq : Real.exp (-χ+ε)+δ<1) |
| (w v : ℂ) (hv : ∀ n : ℕ, FineRegular k χ ε (complexProjection ((standardLift k)^[n] v))) |
| (s : ℝ) : Plane := |
| fineInverse k χ ε δ (complexProjection w) (fineStableCurve k χ ε δ hδ hq v hv s-w) |
|
|
| lemma fine_plaque_linear_error (k χ ε δ : ℝ) (hδ : 0<δ) (hq : Real.exp (-χ+ε)+δ<1) |
| (w v : ℂ) (hv : ∀ n : ℕ, FineRegular k χ ε (complexProjection ((standardLift k)^[n] v))) |
| (hclose : ‖(fineInverse k χ ε δ (complexProjection w)).comp (fineFrame k χ ε δ (complexProjection v))- |
| ContinuousLinearMap.id ℝ Plane‖≤1/4) (s t : ℝ) : |
| |(finePlaqueInChart k χ ε δ hδ hq w v hv s).1- |
| (finePlaqueInChart k χ ε δ hδ hq w v hv t).1-(s-t)|≤(1/4)*|s-t| := by |
| let I := fineInverse k χ ε δ (complexProjection w) |
| let F := fineFrame k χ ε δ (complexProjection v) |
| let u : Plane := (s,fineStableGraph k χ ε δ hδ hq v hv s)-(t,fineStableGraph k χ ε δ hδ hq v hv t) |
| have he : (finePlaqueInChart k χ ε δ hδ hq w v hv s).1- |
| (finePlaqueInChart k χ ε δ hδ hq w v hv t).1-(s-t)= |
| (((I.comp F)-ContinuousLinearMap.id ℝ Plane) u).1 := by |
| change (I (v+F (s,fineStableGraph k χ ε δ hδ hq v hv s)-w)).1- |
| (I (v+F (t,fineStableGraph k χ ε δ hδ hq v hv t)-w)).1-(s-t)=_ |
| simp only [map_sub,map_add,Prod.fst_add,Prod.fst_sub,sub_apply, |
| ContinuousLinearMap.comp_apply,ContinuousLinearMap.id_apply,u] |
| ring |
| rw [he] |
| exact (norm_fst_le _).trans ((ContinuousLinearMap.le_opNorm _ _).trans |
| ((mul_le_mul_of_nonneg_right hclose (norm_nonneg u)).trans |
| (mul_le_mul_of_nonneg_left (fine_graph_pair_bound k χ ε δ hδ hq v hv s t) (by norm_num)))) |
|
|
| lemma fine_plaque_crossing_unique (k χ ε δ : ℝ) (hδ : 0<δ) (hq : Real.exp (-χ+ε)+δ<1) |
| (w v : ℂ) (hv : ∀ n : ℕ, FineRegular k χ ε (complexProjection ((standardLift k)^[n] v))) |
| (hclose : ‖(fineInverse k χ ε δ (complexProjection w)).comp (fineFrame k χ ε δ (complexProjection v))- |
| ContinuousLinearMap.id ℝ Plane‖≤1/4) : |
| Function.Injective (fun s => (finePlaqueInChart k χ ε δ hδ hq w v hv s).1) := |
| (strictMono_of_linear_error (by norm_num) (fine_plaque_linear_error k χ ε δ hδ hq w v hv hclose)).injective |
|
|
| theorem fine_compact_graph_crossing {X : Type*} [TopologicalSpace X] [CompactSpace X] [T2Space X] |
| (k χ ε δ : ℝ) (hδ : 0<δ) (hq : Real.exp (-χ+ε)+δ<1) |
| (w : ℂ) (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} (hr : 0<r) (hr1 : 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) |
| (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) := by |
| let F (x : X) (s : ℝ) := finePlaqueInChart k χ ε δ hδ hq w (v x) (hv x) s |
| have hc0 := continuous_fineStableCurve_family k χ ε δ hδ hq v hvc hv hdata |
| let ι : Icc (-r) r → Icc (-1 : ℝ) 1 := fun s => ⟨s,⟨by linarith [s.property.1],s.property.2.trans hr1⟩⟩ |
| have hι : Continuous ι := continuous_subtype_val.subtype_mk _ |
| have hc : Continuous (fun p : X×Icc (-r) r => F p.1 p.2) := by |
| exact (fineInverse k χ ε δ (complexProjection w)).continuous.comp |
| ((hc0.comp (continuous_fst.prodMk (hι.comp continuous_snd))).sub continuous_const) |
| apply compact_graph_crossing r hr F hc (fun x => fine_plaque_linear_error k χ ε δ hδ hq w (v x) (hv x) (hclose x)) |
| intro x |
| simpa only [F,finePlaqueInChart,fineStableCurve_zero] using hcentre x |
|
|
| end StandardMapEntropy |
|
|
| end |
| section |
| namespace StandardMapEntropy |
| open MeasureTheory Set Filter |
| open scoped Topology NNReal ENNReal |
| open NonlinearStable |
|
|
| theorem fine_small_crossings_coincide (k χ ε δ : ℝ) (hδ : 0<δ) |
| (hq : Real.exp (-χ+ε)+δ<1) (hslow : Real.exp (2*ε)*(Real.exp (-χ+ε)+δ)≤1) |
| (w v v' : ℂ) |
| (hv : ∀ n : ℕ, FineRegular k χ ε (complexProjection ((standardLift k)^[n] v))) |
| (hv' : ∀ n : ℕ, FineRegular k χ ε (complexProjection ((standardLift k)^[n] v'))) |
| {r R : ℝ} (hr : 0<r) (hr1 : r≤1) (hR : 0<R) |
| (hRv : R≤codingRadius k χ ε δ (complexProjection v)) |
| (hsize : (4*fineScale k ε δ*r)/(100*R)+r≤1) |
| (hclose : ‖(fineInverse k χ ε δ (complexProjection w)).comp (fineFrame k χ ε δ (complexProjection v))- |
| ContinuousLinearMap.id ℝ Plane‖≤1/4) |
| {s t s' t' : ℝ} (hs : |s|≤r) (ht : |t|≤r) (hs' : |s'|≤r) (ht' : |t'|≤r) |
| (hmeet : fineStableCurve k χ ε δ hδ hq v hv s=fineStableCurve k χ ε δ hδ hq v' hv' s') |
| (hcross : (finePlaqueInChart k χ ε δ hδ hq w v hv t).1= |
| (finePlaqueInChart k χ ε δ hδ hq w v' hv' t').1) : |
| fineStableCurve k χ ε δ hδ hq v hv t=fineStableCurve k χ ε δ hδ hq v' hv' t' := by |
| let p := fineStableCurve k χ ε δ hδ hq v' hv' t' |
| let p₀ := fineStableCurve k χ ε δ hδ hq v hv s |
| have hscale := fineScale_pos k ε hδ |
| have hq0 : 0≤Real.exp (-χ+ε)+δ := by positivity |
| have htrack (n : ℕ) : ‖(standardLift k)^[n] p-(standardLift k)^[n] p₀‖≤ |
| ((2*fineScale k ε δ)*|t'-s'|)*(Real.exp (-χ+ε)+δ)^n := by |
| dsimp only [p,p₀] |
| rw [hmeet] |
| exact (fineStableCurve_contraction k χ ε δ hδ hq v' hv' (ht'.trans hr1) (hs'.trans hr1) n).trans_eq (by ring) |
| have hdiff : |t'-s'|≤2*r := (abs_sub _ _).trans (by linarith) |
| have hdist (n : ℕ) : ‖fineCoordinate k χ ε δ v p n-fineCoordinate k χ ε δ v p₀ n‖≤ |
| (4*fineScale k ε δ*r)/(100*R) := by |
| have hd := fineCoordinate_tracking_difference k χ ε δ hδ v p p₀ hv (by positivity) hq0 hslow htrack n |
| apply hd.trans |
| calc |
| _ ≤ (4*fineScale k ε δ*r)/(100*codingRadius k χ ε δ (complexProjection v)) := |
| div_le_div_of_nonneg_right (by nlinarith only [hdiff,hscale]) |
| (by positivity [codingRadius_pos k χ ε hδ (complexProjection v)]) |
| _ ≤ _ := div_le_div_of_nonneg_left |
| (mul_nonneg (mul_nonneg (by norm_num) hscale.le) hr.le) (by positivity : 0<100*R) |
| (mul_le_mul_of_nonneg_left hRv (by norm_num)) |
| have hp (n : ℕ) : ‖fineCoordinate k χ ε δ v p n‖≤1 := by |
| have hc : ‖fineCoordinate k χ ε δ v p₀ n‖≤r := |
| (fineStableCoordinate_bound k χ ε δ hδ hq v hv (hs.trans hr1) n).trans hs |
| have hh := norm_add_le (fineCoordinate k χ ε δ v p n-fineCoordinate k χ ε δ v p₀ n) |
| (fineCoordinate k χ ε δ v p₀ n) |
| rw [sub_add_cancel] at hh |
| exact hh.trans ((add_le_add (hdist n) hc).trans hsize) |
| have hu := fineStableCurve_unique k χ ε δ hδ hq v p hv hp |
| let a := (fineCoordinate k χ ε δ v p 0).1 |
| have hca : (finePlaqueInChart k χ ε δ hδ hq w v hv t).1= |
| (finePlaqueInChart k χ ε δ hδ hq w v hv a).1 := by |
| apply hcross.trans |
| change (fineInverse k χ ε δ (complexProjection w) (p-w)).1=_ |
| rw [hu] |
| rfl |
| have hta : t=a := (fine_plaque_crossing_unique k χ ε δ hδ hq w v hv hclose).injOn |
| (s := Icc (-1) 1) (abs_le.mp (ht.trans hr1)) |
| (abs_le.mp ((norm_fst_le _).trans (hp 0))) hca |
| rw [hta] |
| exact hu.symm |
| |
| end StandardMapEntropy |
| |
| end |
| end |
| |
| end OAI |
| |
