| import OAI.Dynamics.StandardMap.EntropyEndpoint |
| import OAI.Dynamics.StandardMap.Entropy.FineCharts |
|
|
| namespace OAI |
|
|
| section |
| section |
| namespace StandardMapEntropy |
| open MeasureTheory Set Filter |
| open scoped Topology ENNReal |
|
|
| lemma standardDerivativeProduct_quantitative (k : ℝ) (hk : 0≤k) : |
| ∃ C : ℝ, 2≤C ∧ ∀ n : ℕ, |
| (∀ z : ℂ, ‖standardDerivativeProduct k (complexProjection z) n‖≤C^n) ∧ |
| (∀ z w : ℂ, ‖standardDerivativeProduct k (complexProjection z) n- |
| standardDerivativeProduct k (complexProjection w) n‖≤C^(3*n)*‖z-w‖) := by |
| let D := 9*growthBase k |
| let L := derivativeVariationBound k |
| let C := 2+D+L |
| have hD : 0≤D := by dsimp [D]; have := growthBase_ge_four k hk; positivity |
| have hL : 0≤L := derivativeVariationBound_nonneg k |
| have hC2 : 2≤C := by dsimp [C]; linarith |
| have hC1 : 1≤C := by linarith |
| have hC0 : 0≤C := by linarith |
| have hDC : D≤C := by dsimp [C]; linarith |
| refine ⟨C,hC2,?_⟩ |
| intro n |
| induction n with |
| | zero => |
| constructor |
| · intro z; simp [standardDerivativeProduct] |
| · intro z w; simp [standardDerivativeProduct] |
| | succ n ih => |
| have hf : ∀ z w : ℂ, ‖(standardLift k)^[n] z-(standardLift k)^[n] w‖≤C^n*‖z-w‖ := |
| norm_sub_le_of_derivative_bound _ _ (hasFDerivAt_standardLift_iterate k n) ih.1 |
| constructor |
| · intro z |
| exact (ContinuousLinearMap.opNorm_comp_le _ _).trans |
| ((mul_le_mul ((standardDerivative_norm_bound k hk _).trans hDC) (ih.1 z) |
| (norm_nonneg _) hC0).trans_eq (by rw [pow_succ]; ring)) |
| · intro z w |
| let Az := standardDerivative k ((standardMap k)^[n] (complexProjection z)) |
| let Aw := standardDerivative k ((standardMap k)^[n] (complexProjection w)) |
| let Pz := standardDerivativeProduct k (complexProjection z) n |
| let Pw := standardDerivativeProduct k (complexProjection w) n |
| have he : Az.comp Pz-Aw.comp Pw=(Az-Aw).comp Pz+Aw.comp (Pz-Pw) := by |
| ext v |
| simp only [sub_apply,add_apply,ContinuousLinearMap.comp_apply,map_sub] |
| abel |
| have hdiff : ‖Az-Aw‖≤L*(C^n*‖z-w‖) := by |
| have hh := standardDerivative_lift_sub_bound k ((standardLift k)^[n] z) ((standardLift k)^[n] w) |
| rw [complexProjection_iterate,complexProjection_iterate] at hh |
| exact hh.trans (mul_le_mul_of_nonneg_left (hf z w) hL) |
| have hn : C^(2*n)≤C^(3*n) := pow_le_pow_right₀ hC1 (by omega) |
| have hsum : L*C^(2*n)+D*C^(3*n)≤C^(3*(n+1)) := by |
| calc |
| _ ≤ (L+D)*C^(3*n) := by nlinarith [mul_le_mul_of_nonneg_left hn hL] |
| _ ≤ C*C^(3*n) := mul_le_mul_of_nonneg_right (by dsimp [C]; linarith) (pow_nonneg hC0 _) |
| _ = C^(3*n+1) := by rw [pow_succ]; ring |
| _ ≤ _ := pow_le_pow_right₀ hC1 (by omega) |
| change ‖Az.comp Pz-Aw.comp Pw‖≤_ |
| rw [he] |
| calc |
| _ ≤ ‖(Az-Aw).comp Pz‖+‖Aw.comp (Pz-Pw)‖ := norm_add_le _ _ |
| _ ≤ ‖Az-Aw‖*‖Pz‖+‖Aw‖*‖Pz-Pw‖ := |
| add_le_add (ContinuousLinearMap.opNorm_comp_le _ _) (ContinuousLinearMap.opNorm_comp_le _ _) |
| _ ≤ (L*(C^n*‖z-w‖))*C^n+D*(C^(3*n)*‖z-w‖) := by |
| apply add_le_add |
| · exact mul_le_mul hdiff (ih.1 z) (norm_nonneg _) (by positivity) |
| · exact mul_le_mul (standardDerivative_norm_bound k hk _) (ih.2 z w) (norm_nonneg _) hD |
| _ = (L*C^(2*n)+D*C^(3*n))*‖z-w‖ := by rw [pow_mul]; ring |
| _ ≤ _ := mul_le_mul_of_nonneg_right hsum (norm_nonneg _) |
|
|
| lemma standardInverseDerivativeProduct_quantitative (k : ℝ) (hk : 0≤k) : |
| ∃ C : ℝ, 2≤C ∧ ∀ n : ℕ, ∀ z w : ℂ, |
| ‖standardInverseDerivativeProduct k (complexProjection z) n- |
| standardInverseDerivativeProduct k (complexProjection w) n‖≤243*C^(3*n)*‖z-w‖ := by |
| obtain ⟨C,hC,h⟩ := standardDerivativeProduct_quantitative k hk |
| refine ⟨C,hC,?_⟩ |
| intro n z w |
| have he : standardInverseDerivativeProduct k (complexProjection z) n- |
| standardInverseDerivativeProduct k (complexProjection w) n = tangentReversal.comp |
| ((standardDerivativeProduct k (complexProjection (tangentReversal z)) n- |
| standardDerivativeProduct k (complexProjection (tangentReversal w)) n).comp tangentReversal) := by |
| rw [standardInverseDerivativeProduct_conjugate,standardInverseDerivativeProduct_conjugate, |
| complexProjection_tangentReversal,complexProjection_tangentReversal] |
| ext v |
| simp only [ContinuousLinearMap.comp_apply,sub_apply,map_sub] |
| rw [he] |
| have hc := tangentReversal_conjugate_norm_le |
| (standardDerivativeProduct k (complexProjection (tangentReversal z)) n- |
| standardDerivativeProduct k (complexProjection (tangentReversal w)) n) |
| have hdist : ‖tangentReversal z-tangentReversal w‖≤3*‖z-w‖ := by |
| rw [←map_sub]; exact tangentReversal_norm_le _ |
| calc |
| _ ≤ 81*‖standardDerivativeProduct k (complexProjection (tangentReversal z)) n- |
| standardDerivativeProduct k (complexProjection (tangentReversal w)) n‖ := hc |
| _ ≤ 81*(C^(3*n)*‖tangentReversal z-tangentReversal w‖) := |
| mul_le_mul_of_nonneg_left ((h n).2 _ _) (by norm_num) |
| _ ≤ 81*(C^(3*n)*(3*‖z-w‖)) := |
| mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left hdist (pow_nonneg (by linarith) _)) (by norm_num) |
| _ = _ := by ring |
|
|
| end StandardMapEntropy |
|
|
| end |
| section |
| namespace StandardMapEntropy |
| open MeasureTheory Set Filter |
| open scoped Topology ENNReal |
|
|
| lemma contracting_line_comparison (A B : ℂ →L[ℝ] ℂ) (hA : PlaneAreaPreserving A) |
| {u v : ℂ} (hv : ‖v‖=1) {a b : ℝ} (ha : ‖A u‖≤a) (hb : ‖B v‖≤b) : |
| |wedge u v|≤a*b+a*‖A-B‖ := by |
| have ha0 : 0≤a := (norm_nonneg _).trans ha |
| have hb0 : 0≤b := (norm_nonneg _).trans hb |
| have he : A v=B v+(A-B) v := by simp only [sub_apply]; abel |
| calc |
| _ = |wedge (A u) (A v)| := congrArg abs (hA u v).symm |
| _ ≤ ‖A u‖*‖A v‖ := abs_wedge_le_norm_mul _ _ |
| _ ≤ a*(b+‖A-B‖) := by |
| apply mul_le_mul ha _ (norm_nonneg _) ha0 |
| calc |
| _ = ‖B v+(A-B) v‖ := congrArg norm he |
| _ ≤ ‖B v‖+‖(A-B) v‖ := norm_add_le _ _ |
| _ ≤ b+‖A-B‖ := add_le_add hb (by simpa only [hv,mul_one] using (A-B).le_opNorm v) |
| _ = _ := by ring |
|
|
| lemma PlaneLyapunov.norm_le_of_energy_le (A : ℕ → ℂ →L[ℝ] ℂ) {χ : ℝ} {v : ℂ} |
| (hsum : Summable (fun n : ℕ => ‖A n v‖^2*Real.exp (2*χ*(n : ℝ)))) |
| {C : ℝ} (hC : 0≤C) (hE : PlaneLyapunov.energy A χ v≤C^2) (n : ℕ) : |
| ‖A n v‖≤C*Real.exp (-χ*(n : ℝ)) := by |
| have ht := (hsum.le_tsum n (fun m _ => mul_nonneg (sq_nonneg _) (Real.exp_pos _).le)).trans hE |
| have he : Real.exp (2*χ*(n : ℝ))=(Real.exp (χ*(n : ℝ)))^2 := by |
| rw [pow_two,←Real.exp_add]; congr 1; ring |
| rw [he] at ht |
| have hh : ‖A n v‖*Real.exp (χ*(n : ℝ))≤C := |
| (sq_le_sq₀ (mul_nonneg (norm_nonneg _) (Real.exp_pos _).le) hC).mp (by nlinarith) |
| have hh' := mul_le_mul_of_nonneg_right hh (Real.exp_pos (-χ*(n : ℝ))).le |
| have hc : Real.exp (χ*(n : ℝ))*Real.exp (-χ*(n : ℝ))=1 := by |
| rw [←Real.exp_add]; convert Real.exp_zero using 1; ring_nf |
| simpa only [mul_assoc,hc,mul_one] using hh' |
|
|
| lemma unit_projective_image_bound (A : ℂ →L[ℝ] ℂ) {u v : ℂ} |
| (hu : ‖u‖=1) (hv : ‖v‖=1) : |
| ‖A v‖≤‖A u‖+‖A‖*|wedge u v| := by |
| have hd : |dot u v|≤1 := by simpa only [hu,hv,one_mul] using abs_dot_le_norm_mul u v |
| have hj : ‖A (quarterTurn u)‖≤‖A‖ := by |
| simpa only [norm_quarterTurn,hu,mul_one] using A.le_opNorm (quarterTurn u) |
| calc |
| _ = ‖dot u v • A u+wedge u v • A (quarterTurn u)‖ := by |
| conv_lhs => rw [unit_expansion u v hu,map_add,map_smul,map_smul] |
| _ ≤ |dot u v| *‖A u‖+|wedge u v| *‖A (quarterTurn u)‖ := by |
| simpa only [norm_smul,Real.norm_eq_abs] using norm_add_le (dot u v • A u) (wedge u v • A (quarterTurn u)) |
| _ ≤ 1*‖A u‖+|wedge u v| *‖A‖ := |
| add_le_add (mul_le_mul_of_nonneg_right hd (norm_nonneg _)) (mul_le_mul_of_nonneg_left hj (abs_nonneg _)) |
| _ = _ := by ring |
|
|
| lemma unit_image_norm_difference (A B : ℂ →L[ℝ] ℂ) {u v : ℂ} |
| (hu : ‖u‖=1) (hv : ‖v‖=1) : |
| |‖A u‖-‖B v‖|≤‖A‖*|wedge u v|+‖A-B‖ := by |
| have h1 := unit_projective_image_bound A hu hv |
| have h2 := unit_projective_image_bound A hv hu |
| rw [wedge_swap v u,abs_neg] at h2 |
| have h3 : |‖A v‖-‖B v‖|≤‖A-B‖ := |
| (abs_norm_sub_norm_le _ _).trans (by simpa only [sub_apply,hv,mul_one] using (A-B).le_opNorm v) |
| have h4 := abs_add_le (‖A u‖-‖A v‖) (‖A v‖-‖B v‖) |
| have h5 : |‖A u‖-‖A v‖|≤‖A‖*|wedge u v| := abs_le.mpr ⟨by linarith,by linarith⟩ |
| have he : (‖A u‖-‖A v‖)+(‖A v‖-‖B v‖)=‖A u‖-‖B v‖ := by ring |
| rw [he] at h4 |
| exact h4.trans (add_le_add h5 h3) |
|
|
| lemma log_difference_of_lower {a b D : ℝ} (hD : 0<D) (ha : 1≤D*a) (hb : 1≤D*b) : |
| |Real.log a-Real.log b|≤D*|a-b| := by |
| have ha0 : 0<a := by nlinarith |
| have hb0 : 0<b := by nlinarith |
| have hab := Real.log_le_sub_one_of_pos (div_pos ha0 hb0) |
| have hba := Real.log_le_sub_one_of_pos (div_pos hb0 ha0) |
| rw [Real.log_div ha0.ne' hb0.ne'] at hab |
| rw [Real.log_div hb0.ne' ha0.ne'] at hba |
| have hab' : a/b-1≤D*|a-b| := by |
| apply (sub_le_iff_le_add).mpr |
| apply (div_le_iff₀ hb0).mpr |
| nlinarith [le_abs_self (a-b),mul_le_mul_of_nonneg_left hb (abs_nonneg (a-b))] |
| have hba' : b/a-1≤D*|a-b| := by |
| apply (sub_le_iff_le_add).mpr |
| apply (div_le_iff₀ ha0).mpr |
| nlinarith [neg_le_abs (a-b),mul_le_mul_of_nonneg_left ha (abs_nonneg (a-b))] |
| exact abs_le.mpr ⟨by linarith,hab.trans hab'⟩ |
|
|
| lemma area_unit_log_image_difference (A B : ℂ →L[ℝ] ℂ) |
| (hA : PlaneAreaPreserving A) (hB : PlaneAreaPreserving B) {u v : ℂ} |
| (hu : ‖u‖=1) (hv : ‖v‖=1) {D : ℝ} (hD : 0<D) (hDA : ‖A‖≤D) (hDB : ‖B‖≤D) : |
| |Real.log ‖A u‖-Real.log ‖B v‖|≤D^2*|wedge u v|+D*‖A-B‖ := by |
| have ha : 1≤D*‖A u‖ := by |
| calc |
| _ = ‖u‖ := hu.symm |
| _ ≤ ‖A‖*‖A u‖ := hA.norm_lower u |
| _ ≤ _ := mul_le_mul_of_nonneg_right hDA (norm_nonneg _) |
| have hb : 1≤D*‖B v‖ := by |
| calc |
| _ = ‖v‖ := hv.symm |
| _ ≤ ‖B‖*‖B v‖ := hB.norm_lower v |
| _ ≤ _ := mul_le_mul_of_nonneg_right hDB (norm_nonneg _) |
| calc |
| _ ≤ D*|‖A u‖-‖B v‖| := log_difference_of_lower hD ha hb |
| _ ≤ D*(‖A‖*|wedge u v|+‖A-B‖) := mul_le_mul_of_nonneg_left (unit_image_norm_difference A B hu hv) hD.le |
| _ ≤ D*(D*|wedge u v|+‖A-B‖) := mul_le_mul_of_nonneg_left |
| (add_le_add (mul_le_mul_of_nonneg_right hDA (abs_nonneg _)) le_rfl) hD.le |
| _ = _ := by ring |
|
|
| noncomputable def unstableLogJacobian (k : ℝ) (z : Torus) : ℝ := |
| Real.log ‖standardDerivative k z (unstableVector k z)‖ |
| lemma measurable_unstableLogJacobian (k : ℝ) : Measurable (unstableLogJacobian k) := |
| (((continuous_fst.clm_apply continuous_snd : |
| Continuous (fun p : (ℂ →L[ℝ] ℂ) × ℂ => p.1 p.2)).measurable.comp |
| ((continuous_standardDerivative k).measurable.prodMk (measurable_unstableVector k))).norm).log |
|
|
| lemma unstableLogJacobian_difference (k : ℝ) (hk : 0≤k) (z w : ℂ) : |
| |unstableLogJacobian k (complexProjection z)-unstableLogJacobian k (complexProjection w)|≤ |
| (9*growthBase k)^2*|wedge (unstableVector k (complexProjection z)) (unstableVector k (complexProjection w))|+ |
| (9*growthBase k)*derivativeVariationBound k*‖z-w‖ := by |
| have hD : 0<9*growthBase k := by have := growthBase_ge_four k hk; positivity |
| exact (area_unit_log_image_difference _ _ (standardDerivative_area k _) (standardDerivative_area k _) |
| (norm_unstableVector k _) (norm_unstableVector k _) hD |
| (standardDerivative_norm_bound k hk _) (standardDerivative_norm_bound k hk _)).trans |
| (add_le_add le_rfl ((mul_le_mul_of_nonneg_left (standardDerivative_lift_sub_bound k z w) hD.le).trans_eq (by ring))) |
|
|
| end StandardMapEntropy |
|
|
| end |
| section |
| namespace StandardMapEntropy |
| open Finset |
|
|
| lemma sum_of_backward_contraction (a : ℕ → ℝ) (ha : ∀ n,0≤a n) {q : ℝ} |
| (hq0 : 0≤q) (hq1 : q<1) (N : ℕ) (hstep : ∀ j : ℕ,j<N → a j≤q*a (j+1)) : |
| (∑ j ∈ Finset.range N,a j)≤a N/(1-q) := by |
| have hh : (∑ j ∈ Finset.range N,a j)≤q*(∑ j ∈ Finset.range N,a (j+1)) := by |
| rw [Finset.mul_sum] |
| exact Finset.sum_le_sum fun j hj => hstep j (Finset.mem_range.mp hj) |
| have he : (∑ j ∈ Finset.range N,a (j+1))+a 0=(∑ j ∈ Finset.range N,a j)+a N := by |
| rw [←Finset.sum_range_succ',←Finset.sum_range_succ] |
| apply (le_div_iff₀ (sub_pos.mpr hq1)).mpr |
| have hlast := mul_le_mul_of_nonneg_right hq1.le (ha N) |
| have hfirst := mul_nonneg hq0 (ha 0) |
| nlinarith |
|
|
| namespace NonlinearStable |
| open Set Filter MeasureTheory |
| open scoped Topology NNReal ENNReal |
| variable {ℓ δ : ℝ≥0} |
|
|
| @[simp] lemma Recurrence.step_zero (r : Recurrence ℓ δ) (n : ℕ) : r.step n 0=0 := by |
| simp [Recurrence.step,r.zero] |
| @[simp] lemma Recurrence.orbit_zero (r : Recurrence ℓ δ) (n : ℕ) : r.orbit 0 n=0 := by |
| induction n with |
| | zero => rfl |
| | succ n ih => simp only [Recurrence.orbit,ih,r.step_zero] |
|
|
| lemma Recurrence.orbit_cone (r : Recurrence ℓ δ) (h : ℓ+δ<1) {v : Plane} |
| (hv : |v.1|≤|v.2|) (n : ℕ) : |
| |(r.orbit v n).1|≤|(r.orbit v n).2| ∧ |
| |(r.orbit v n).2|≤((ℓ+δ : ℝ≥0) : ℝ)*|(r.orbit v (n+1)).2| := by |
| have hcone (m : ℕ) : |(r.orbit v m).1|≤|(r.orbit v m).2| := by |
| induction m with |
| | zero => exact hv |
| | succ m ih => |
| simpa only [Recurrence.orbit,r.step_zero,Prod.fst_zero,Prod.snd_zero,sub_zero] using |
| (r.unstable_cone_step h m (r.orbit v m) 0 (by simpa only [Prod.fst_zero,Prod.snd_zero,sub_zero] using ih)).1 |
| refine ⟨hcone n,?_⟩ |
| simpa only [Recurrence.orbit,r.step_zero,Prod.snd_zero,sub_zero] using |
| (r.unstable_cone_step h n (r.orbit v n) 0 (by simpa only [Prod.fst_zero,Prod.snd_zero,sub_zero] using hcone n)).2 |
| lemma Recurrence.orbit_norm_eq_snd (r : Recurrence ℓ δ) (h : ℓ+δ<1) {v : Plane} |
| (hv : |v.1|≤|v.2|) (n : ℕ) : ‖r.orbit v n‖=|(r.orbit v n).2| := |
| max_eq_right (r.orbit_cone h hv n).1 |
| lemma Recurrence.orbit_snd_ne_zero (r : Recurrence ℓ δ) (h : ℓ+δ<1) {v : Plane} |
| (hv : |v.1|≤|v.2|) (hv0 : v.2≠0) (n : ℕ) : (r.orbit v n).2≠0 := by |
| induction n with |
| | zero => exact hv0 |
| | succ n ih => |
| intro he |
| have hc := (r.orbit_cone h hv n).2 |
| rw [he,abs_zero,mul_zero] at hc |
| exact (abs_pos.mpr ih).not_ge hc |
|
|
| lemma scalar_log_quadratic_error {b y e d : ℝ} (hb : 1≤|b|) (hy : y≠0) |
| (hd0 : 0≤d) (hd1 : d≤1/2) (hysize : |y|≤1) (he : |e|≤d*|y|^2) : |
| abs (Real.log |b*y+e|-Real.log |b|-Real.log |y|) ≤2*d*|y| := by |
| have hyp : 0 < |y| := abs_pos.mpr hy |
| have hbp : 0 < |b| := lt_of_lt_of_le zero_lt_one hb |
| have hsmall : |e|≤|y|/2 := by |
| have hsq : |y|^2≤|y| := by nlinarith [abs_nonneg y] |
| have hle := mul_le_mul_of_nonneg_left hsq hd0 |
| have hle' := mul_le_mul_of_nonneg_right hd1 (abs_nonneg y) |
| nlinarith |
| have hlower : |y|/2≤|b*y+e| := by |
| have hh : |b*y|≤|b*y+e|+|e| := by |
| have hh := abs_sub (b*y+e) e |
| simpa only [add_sub_cancel_right] using hh |
| rw [abs_mul] at hh |
| have hb' := mul_le_mul_of_nonneg_right hb (abs_nonneg y) |
| nlinarith |
| have hD : 0<2/|y| := div_pos (by norm_num) hyp |
| have hx : 1≤(2/|y|)*|b*y+e| := by |
| have hc := mul_le_mul_of_nonneg_left hlower hD.le |
| have heq : (2/|y|)*(|y|/2)=1 := by field_simp |
| rwa [heq] at hc |
| have hb' : 1≤(2/|y|)*|b*y| := by |
| rw [abs_mul] |
| have heq : (2/|y|)*(|b| *|y|)=2*|b| := by field_simp |
| rw [heq]; linarith |
| have hl := log_difference_of_lower hD hx hb' |
| have hnorm : abs (|b*y+e|-|b*y|)≤|e| := by |
| simpa only [add_sub_cancel_left] using abs_abs_sub_abs_le_abs_sub (b*y+e) (b*y) |
| have hh := hl.trans (mul_le_mul_of_nonneg_left (hnorm.trans he) hD.le) |
| rw [abs_mul,Real.log_mul hbp.ne' hyp.ne'] at hh |
| have hright : (2/|y|)*(d*|y|^2)=2*d*|y| := by field_simp |
| rw [hright] at hh |
| convert hh using 1; congr 1; ring |
|
|
| theorem Recurrence.finite_cone_log_distortion (r : Recurrence ℓ δ) (h : ℓ+δ<1) |
| (hδ : (δ : ℝ)≤1/2) |
| (hquad : ∀ n : ℕ,∀ v : Plane,‖v‖≤1 → ‖r.remainder n v‖≤(δ : ℝ)*‖v‖^2) |
| {v : Plane} (hv : |v.1|≤|v.2|) (hv0 : v.2≠0) (N : ℕ) (htrap : v∈r.trappedBox N) : |
| abs (Real.log |(r.orbit v N).2|-Real.log |v.2|- |
| ∑ j ∈ Finset.range N,Real.log |r.b j|) ≤2*(δ : ℝ)/(1-((ℓ+δ : ℝ≥0) : ℝ)) := by |
| let y := fun j : ℕ => |(r.orbit v j).2| |
| have hstep (j : ℕ) (hj : j<N) : |
| abs ((Real.log (y (j+1))-Real.log (y j))-Real.log |r.b j|) ≤2*(δ : ℝ)*y j := by |
| have hy := r.orbit_snd_ne_zero h hv hv0 j |
| have hysize : |(r.orbit v j).2|≤1 := by |
| rw [←r.orbit_norm_eq_snd h hv j] |
| exact htrap j hj.le |
| have herr : |(r.remainder j (r.orbit v j)).2|≤(δ : ℝ)*|(r.orbit v j).2|^2 := by |
| have hh := (norm_snd_le (r.remainder j (r.orbit v j))).trans (hquad j _ (htrap j hj.le)) |
| simpa only [Real.norm_eq_abs,r.orbit_norm_eq_snd h hv j] using hh |
| have hh := scalar_log_quadratic_error (r.unstable_size_ge_one h j) hy δ.coe_nonneg hδ hysize herr |
| change abs (Real.log |r.b j*(r.orbit v j).2+(r.remainder j (r.orbit v j)).2|- |
| Real.log |(r.orbit v j).2|-Real.log |r.b j|) ≤_ |
| convert hh using 1; congr 1; ring |
| have hsum := sum_of_backward_contraction y (fun j => abs_nonneg _) (ℓ+δ).coe_nonneg |
| (NNReal.coe_lt_coe.mpr h) N (fun j _ => (r.orbit_cone h hv j).2) |
| have hlast : y N≤1 := by |
| dsimp [y]; rw [←r.orbit_norm_eq_snd h hv N]; exact htrap N le_rfl |
| have hsum' : (∑ j ∈ Finset.range N,y j)≤1/(1-((ℓ+δ : ℝ≥0) : ℝ)) := |
| hsum.trans (div_le_div_of_nonneg_right hlast (sub_nonneg.mpr (NNReal.coe_le_coe.mpr h.le))) |
| have htel : (∑ j ∈ Finset.range N,((Real.log (y (j+1))-Real.log (y j))-Real.log |r.b j|))= |
| Real.log (y N)-Real.log (y 0)-(∑ j ∈ Finset.range N,Real.log |r.b j|) := by |
| rw [Finset.sum_sub_distrib] |
| congr 1 |
| exact Finset.sum_range_sub (fun j => Real.log (y j)) N |
| calc |
| _ = |∑ j ∈ Finset.range N,((Real.log (y (j+1))-Real.log (y j))-Real.log |r.b j|)| := congrArg abs htel.symm |
| _ ≤ ∑ j ∈ Finset.range N,abs ((Real.log (y (j+1))-Real.log (y j))-Real.log |r.b j|) := Finset.abs_sum_le_sum_abs _ _ |
| _ ≤ ∑ j ∈ Finset.range N,2*(δ : ℝ)*y j := Finset.sum_le_sum fun j hj => hstep j (Finset.mem_range.mp hj) |
| _ = (2*(δ : ℝ))*(∑ j ∈ Finset.range N,y j) := (Finset.mul_sum _ _ _).symm |
| _ ≤ (2*(δ : ℝ))*(1/(1-((ℓ+δ : ℝ≥0) : ℝ))) := mul_le_mul_of_nonneg_left hsum' (by positivity) |
| _ = _ := by ring |
|
|
| end NonlinearStable |
| end StandardMapEntropy |
|
|
| end |
| section |
| namespace StandardMapEntropy |
| open MeasureTheory Set Filter |
| open scoped Topology NNReal ENNReal |
| open NonlinearStable |
|
|
| noncomputable def fineRemainderDerivative (k χ ε δ : ℝ) (z : ℂ) : RealPlane → RealPlane →L[ℝ] RealPlane := |
| flatRemainderDerivative k z (fineFrame k χ ε δ (complexProjection z)) |
| (fineInverse k χ ε δ (standardMap k (complexProjection z))) |
| lemma hasFDerivAt_fineRemainder (k χ ε δ : ℝ) (z : ℂ) (v : RealPlane) : |
| HasFDerivAt (fineRemainder k χ ε δ z) (fineRemainderDerivative k χ ε δ z v) v := |
| hasFDerivAt_flatRemainder _ _ _ _ _ |
| @[simp] lemma fineRemainderDerivative_zero (k χ ε δ : ℝ) (z : ℂ) : fineRemainderDerivative k χ ε δ z 0=0 := |
| flatRemainderDerivative_zero _ _ _ _ |
| lemma fineRemainderDerivative_lipschitz (k χ ε : ℝ) {δ : ℝ} (hδ : 0<δ) (z : ℂ) |
| (hz : FineRegular k χ ε (complexProjection z)) |
| (hn : FineRegular k χ ε (standardMap k (complexProjection z))) (v w : RealPlane) : |
| ‖fineRemainderDerivative k χ ε δ z v-fineRemainderDerivative k χ ε δ z w‖≤δ*‖v-w‖ := |
| (flatRemainderDerivative_lipschitz k z _ _ v w).trans |
| (mul_le_mul_of_nonneg_right (fine_nonlinearity_coefficient k χ ε hδ _ hz hn) (norm_nonneg _)) |
| lemma fineRemainderDerivative_bound (k χ ε : ℝ) {δ : ℝ} (hδ : 0<δ) (z : ℂ) |
| (hz : FineRegular k χ ε (complexProjection z)) |
| (hn : FineRegular k χ ε (standardMap k (complexProjection z))) |
| {v : RealPlane} (hv : ‖v‖≤1) : ‖fineRemainderDerivative k χ ε δ z v‖≤δ := by |
| have hh := fineRemainderDerivative_lipschitz k χ ε hδ z hz hn v 0 |
| simp only [fineRemainderDerivative_zero,sub_zero] at hh |
| exact hh.trans ((mul_le_mul_of_nonneg_left hv hδ.le).trans_eq (mul_one _)) |
| lemma fineRemainder_taylor (k χ ε : ℝ) {δ : ℝ} (hδ : 0<δ) (z : ℂ) |
| (hz : FineRegular k χ ε (complexProjection z)) |
| (hn : FineRegular k χ ε (standardMap k (complexProjection z))) (v w : RealPlane) : |
| ‖fineRemainder k χ ε δ z w-fineRemainder k χ ε δ z v- |
| fineRemainderDerivative k χ ε δ z v (w-v)‖≤δ*‖w-v‖^2 := |
| quadratic_taylor_of_derivative_lipschitz _ _ (hasFDerivAt_fineRemainder k χ ε δ z) |
| hδ.le (fineRemainderDerivative_lipschitz k χ ε hδ z hz hn) v w |
| lemma fineRemainder_quadratic (k χ ε : ℝ) {δ : ℝ} (hδ : 0<δ) (z : ℂ) |
| (hz : FineRegular k χ ε (complexProjection z)) |
| (hn : FineRegular k χ ε (standardMap k (complexProjection z))) (v : RealPlane) : |
| ‖fineRemainder k χ ε δ z v‖≤δ*‖v‖^2 := by |
| simpa only [fineRemainder_zero,fineRemainderDerivative_zero,zero_apply,sub_zero] using |
| fineRemainder_taylor k χ ε hδ z hz hn 0 v |
|
|
| lemma fineTrapped_orbit (k χ ε δ : ℝ) (hδ : 0<δ) (w : ℂ) |
| (hw : ∀ n : ℕ, FineRegular k χ ε (complexProjection ((standardLift k)^[n] w))) |
| {N : ℕ} {v : RealPlane} (hv : v∈fineTrapped k χ ε δ w N) {j : ℕ} (hj : j≤N) : |
| (fineLocalRecurrence k χ ε δ hδ w hw).extend.orbit v j= |
| fineCoordinate k χ ε δ w (w+fineFrame k χ ε δ (complexProjection w) v) j := by |
| induction j with |
| | zero => exact (fineCoordinate_initial k χ ε hδ w (hw 0) v).symm |
| | succ j ih => |
| rw [Recurrence.orbit,ih (by omega),extend_step_eq _ j (hv j (by omega)),fineCoordinate_step] |
|
|
| theorem fineTrapped_log_distortion (k χ ε δ : ℝ) (hδ : 0<δ) (hδ' : δ≤1/2) |
| (hq : Real.exp (-χ+ε)+δ<1) (w : ℂ) |
| (hw : ∀ n : ℕ, FineRegular k χ ε (complexProjection ((standardLift k)^[n] w))) |
| {v : RealPlane} (hv : |v.1|≤|v.2|) (hv0 : v.2≠0) (N : ℕ) |
| (htrap : v∈fineTrapped k χ ε δ w N) : |
| abs (Real.log |(fineCoordinate k χ ε δ w (w+fineFrame k χ ε δ (complexProjection w) v) N).2|- |
| Real.log |v.2|-(∑ j ∈ Finset.range N,Real.log |fineUnstableMultiplier k χ ε δ |
| (complexProjection ((standardLift k)^[j] w))|))≤2*δ/(1-(Real.exp (-χ+ε)+δ)) := by |
| let r := fineLocalRecurrence k χ ε δ hδ w hw |
| have hquad (n : ℕ) (u : RealPlane) (hu : ‖u‖≤1) : ‖r.extend.remainder n u‖≤δ*‖u‖^2 := by |
| change ‖fineRemainder k χ ε δ ((standardLift k)^[n] w) (planeClip u)‖≤_ |
| rw [planeClip_eq hu] |
| exact fineRemainder_quadratic k χ ε hδ _ (hw n) (fine_regular_lift_next k χ ε w hw n) u |
| have hh := r.extend.finite_cone_log_distortion hq hδ' hquad hv hv0 N |
| (fineTrapped_subset k χ ε δ hδ w hw N htrap) |
| rw [fineTrapped_orbit k χ ε δ hδ w hw htrap le_rfl] at hh |
| exact hh |
|
|
| end StandardMapEntropy |
|
|
| end |
| end |
|
|
| end OAI |
|
|
