| import OAI.Dynamics.StandardMap.EntropyEndpoint |
| import OAI.Dynamics.StandardMap.Entropy.SupportEntropy |
|
|
| namespace OAI |
|
|
| section |
| section |
| namespace StandardMapEntropy |
| open MeasureTheory Set Filter |
| open scoped Topology NNReal ENNReal |
|
|
| noncomputable def fineScale (k ε δ : ℝ) : ℝ := δ/(8*(1+derivativeVariationBound k)*Real.exp (2*ε)) |
| noncomputable def fineSize (k χ ε δ : ℝ) (z : Torus) : ℝ := fineScale k ε δ*chartRadius k χ ε z |
| noncomputable def fineFrame (k χ ε δ : ℝ) (z : Torus) : RealPlane →L[ℝ] ℂ := |
| flatFrame (fineSize k χ ε δ z) (lyapunovFrame k χ z) |
| noncomputable def fineInverse (k χ ε δ : ℝ) (z : Torus) : ℂ →L[ℝ] RealPlane := |
| flatInverse (fineSize k χ ε δ z) (lyapunovCoframe k χ z) |
| noncomputable def fineStableMultiplier (k χ ε δ : ℝ) (z : Torus) : ℝ := |
| fineSize k χ ε δ z/fineSize k χ ε δ (standardMap k z)*stableMultiplier k χ z |
| noncomputable def fineUnstableMultiplier (k χ ε δ : ℝ) (z : Torus) : ℝ := |
| fineSize k χ ε δ z/fineSize k χ ε δ (standardMap k z)*unstableMultiplier k χ z |
| lemma fineScale_pos (k ε : ℝ) {δ : ℝ} (hδ : 0<δ) : 0<fineScale k ε δ := by |
| unfold fineScale |
| exact div_pos hδ (mul_pos (mul_pos (by norm_num) (by linarith [derivativeVariationBound_nonneg k])) (Real.exp_pos _)) |
| lemma fineSize_pos (k χ ε : ℝ) {δ : ℝ} (hδ : 0<δ) (z : Torus) : 0<fineSize k χ ε δ z := |
| mul_pos (fineScale_pos k ε hδ) (chartRadius_pos k χ ε z) |
| lemma fineSize_le_scale (k χ ε : ℝ) {δ : ℝ} (hδ : 0<δ) (z : Torus) : fineSize k χ ε δ z≤fineScale k ε δ := |
| (mul_le_mul_of_nonneg_left (chartRadius_le_one k χ ε z) (fineScale_pos k ε hδ).le).trans_eq (mul_one _) |
| lemma fineFrame_norm_le (k χ ε : ℝ) {δ : ℝ} (hδ : 0<δ) (z : Torus) : |
| ‖fineFrame k χ ε δ z‖≤2*fineSize k χ ε δ z := |
| flatFrame_norm_le (fineSize_pos k χ ε hδ z).le (lyapunovFrame_norm_le k χ z) |
| lemma fineInverse_frame (k χ ε : ℝ) {δ : ℝ} (hδ : 0<δ) (z : Torus) |
| (hw : wedge (stableVector k z) (unstableVector k z)≠0) (v : RealPlane) : |
| fineInverse k χ ε δ z (fineFrame k χ ε δ z v)=v := |
| flatInverse_frame (ne_of_gt (fineSize_pos k χ ε hδ z)) (lyapunovCoframe_frame k χ z hw) v |
| lemma fineFrame_inverse (k χ ε : ℝ) {δ : ℝ} (hδ : 0<δ) (z : Torus) |
| (hw : wedge (stableVector k z) (unstableVector k z)≠0) (v : ℂ) : |
| fineFrame k χ ε δ z (fineInverse k χ ε δ z v)=v := |
| flatFrame_inverse (ne_of_gt (fineSize_pos k χ ε hδ z)) (lyapunovFrame_coframe k χ z hw) v |
|
|
| def FineRegular (k χ ε : ℝ) (z : Torus) : Prop := |
| wedge (stableVector k z) (unstableVector k z)≠0 ∧ |
| (lyapunovCoframe k χ (standardMap k z)).comp ((standardDerivative k z).comp (lyapunovFrame k χ z)) = |
| diagonalPlane (stableMultiplier k χ z) (unstableMultiplier k χ z) ∧ |
| |stableMultiplier k χ z|≤Real.exp (-χ) ∧ Real.exp χ≤|unstableMultiplier k χ z| ∧ |
| chartRadius k χ ε z*‖lyapunovCoframe k χ z‖≤1 ∧ |
| Real.exp (-ε)*chartRadius k χ ε z≤chartRadius k χ ε (standardMap k z) ∧ |
| Real.exp (-ε)*chartRadius k χ ε (standardMap k z)≤chartRadius k χ ε z |
| lemma ae_fineRegular (k : ℝ) (hk : 0≤k) {χ ε : ℝ} (hχ : 0<χ) (hε : 0<ε) : |
| ∀ᵐ z ∂area.restrict (spectralGapRegion k hk χ), FineRegular k χ ε z := by |
| filter_upwards [ae_restrict_mem (measurableSet_spectralGapRegion k hk χ), |
| (ae_transverseLines k hk).filter_mono Measure.absolutelyContinuous_restrict.ae_le, |
| ae_lyapunov_normal_form k hk χ hχ.le,ae_stableMultiplier_bound k hk χ hχ.le, |
| ae_unstableMultiplier_bound k hk χ hχ.le,ae_chartRadius_control k hk χ hχ.le ε hε] |
| with z hz ht hn hs hu hr |
| exact ⟨ht (hχ.trans hz),hn,hs,hu,hr⟩ |
| lemma ae_fineRegular_all_iterates (k : ℝ) (hk : 0≤k) {χ ε : ℝ} (hχ : 0<χ) (hε : 0<ε) : |
| ∀ᵐ z ∂area.restrict (spectralGapRegion k hk χ), |
| ∀ n : ℕ, FineRegular k χ ε ((standardMap k)^[n] z) ∧ FineRegular k χ ε ((inverseMap k)^[n] z) := by |
| rw [ae_all_iff] |
| intro n |
| exact (((measurePreserving_spectralGapRegion k hk χ).iterate n).quasiMeasurePreserving.ae |
| (ae_fineRegular k hk hχ hε)).and |
| (((measurePreserving_inverse_spectralGapRegion k hk χ).iterate n).quasiMeasurePreserving.ae |
| (ae_fineRegular k hk hχ hε)) |
| lemma fineSize_ratio (k χ ε : ℝ) {δ : ℝ} (hδ : 0<δ) (z w : Torus) : |
| fineSize k χ ε δ z/fineSize k χ ε δ w=chartRadius k χ ε z/chartRadius k χ ε w := by |
| unfold fineSize |
| rw [mul_div_mul_left _ _ (ne_of_gt (fineScale_pos k ε hδ))] |
| lemma fineStableMultiplier_bound (k χ ε : ℝ) {δ : ℝ} (hδ : 0<δ) (z : Torus) |
| (hz : FineRegular k χ ε z) : |fineStableMultiplier k χ ε δ z|≤Real.exp (-χ+ε) := by |
| have hr := slow_radius_ratio (chartRadius_pos k χ ε z) (chartRadius_pos k χ ε (standardMap k z)) hz.2.2.2.2.2.1 |
| have he : Real.exp ε*Real.exp (-χ)=Real.exp (-χ+ε) := by rw [←Real.exp_add,add_comm] |
| unfold fineStableMultiplier |
| rw [abs_mul,abs_of_pos (div_pos (fineSize_pos k χ ε hδ z) (fineSize_pos k χ ε hδ _)),fineSize_ratio k χ ε hδ] |
| exact (mul_le_mul hr hz.2.2.1 (abs_nonneg _) (Real.exp_pos _).le).trans_eq he |
| lemma fineUnstableMultiplier_ne (k χ ε : ℝ) {δ : ℝ} (hδ : 0<δ) (z : Torus) |
| (hz : FineRegular k χ ε z) : fineUnstableMultiplier k χ ε δ z≠0 := by |
| have hb : unstableMultiplier k χ z≠0 := abs_pos.mp ((Real.exp_pos χ).trans_le hz.2.2.2.1) |
| exact mul_ne_zero (div_ne_zero (ne_of_gt (fineSize_pos k χ ε hδ z)) (ne_of_gt (fineSize_pos k χ ε hδ _))) hb |
| lemma fineUnstableMultiplier_inverse_bound (k χ ε : ℝ) {δ : ℝ} (hδ : 0<δ) (z : Torus) |
| (hz : FineRegular k χ ε z) : |(fineUnstableMultiplier k χ ε δ z)⁻¹|≤Real.exp (-χ+ε) := by |
| have hr := slow_radius_ratio (chartRadius_pos k χ ε (standardMap k z)) (chartRadius_pos k χ ε z) hz.2.2.2.2.2.2 |
| have hb : |unstableMultiplier k χ z|⁻¹≤Real.exp (-χ) := by |
| rw [Real.exp_neg] |
| exact inv_anti₀ (Real.exp_pos χ) hz.2.2.2.1 |
| have he : Real.exp ε*Real.exp (-χ)=Real.exp (-χ+ε) := by rw [←Real.exp_add,add_comm] |
| unfold fineUnstableMultiplier |
| rw [abs_inv,abs_mul,abs_of_pos (div_pos (fineSize_pos k χ ε hδ z) (fineSize_pos k χ ε hδ _)), |
| mul_inv,inv_div,fineSize_ratio k χ ε hδ] |
| exact (mul_le_mul hr hb (inv_nonneg.mpr (abs_nonneg _)) (Real.exp_pos _).le).trans_eq he |
|
|
| end StandardMapEntropy |
|
|
| end |
| section |
| namespace StandardMapEntropy |
| open MeasureTheory Set Filter |
| open scoped Topology ENNReal |
| noncomputable def codingScale (k ε δ : ℝ) : ℝ := min 1 (fineScale k ε δ)/(100*(1+|9*growthBase k|)) |
| noncomputable def codingRadius (k χ ε δ : ℝ) (z : Torus) : ℝ := |
| codingScale k ε δ*(chartRadius k χ ε z)^2 |
| lemma codingScale_pos (k ε : ℝ) {δ : ℝ} (hδ : 0<δ) : 0<codingScale k ε δ := by |
| unfold codingScale |
| exact div_pos (lt_min zero_lt_one (fineScale_pos k ε hδ)) (by positivity) |
| lemma codingScale_le (k ε : ℝ) {δ : ℝ} (hδ : 0<δ) : |
| codingScale k ε δ≤fineScale k ε δ/100 := by |
| have hη := fineScale_pos k ε hδ |
| unfold codingScale |
| calc |
| _ ≤ fineScale k ε δ/(100*(1+|9*growthBase k|)) := div_le_div_of_nonneg_right (min_le_right _ _) (by positivity) |
| _ ≤ _ := div_le_div_of_nonneg_left hη.le (by norm_num) (by nlinarith [abs_nonneg (9*growthBase k)]) |
| lemma codingScale_global (k ε δ : ℝ) : |
| codingScale k ε δ*(1+|9*growthBase k|)≤1/100 := by |
| unfold codingScale |
| have hd : 0<(1+|9*growthBase k|) := by positivity |
| have he : min 1 (fineScale k ε δ)/(100*(1+|9*growthBase k|))*(1+|9*growthBase k|)=min 1 (fineScale k ε δ)/100 := by |
| field_simp |
| rw [he] |
| exact div_le_div_of_nonneg_right (min_le_left _ _) (by norm_num) |
| lemma codingRadius_pos (k χ ε : ℝ) {δ : ℝ} (hδ : 0<δ) (z : Torus) : 0<codingRadius k χ ε δ z := |
| mul_pos (codingScale_pos k ε hδ) (sq_pos_of_pos (chartRadius_pos k χ ε z)) |
| lemma codingRadius_small (k χ ε : ℝ) {δ : ℝ} (hδ : 0<δ) (z : Torus) : |
| (1+|9*growthBase k|)*codingRadius k χ ε δ z≤1/100 := by |
| have hr : (chartRadius k χ ε z)^2≤1 := by |
| have h0 := (chartRadius_pos k χ ε z).le |
| have h1 := chartRadius_le_one k χ ε z |
| nlinarith |
| have hh := mul_le_mul_of_nonneg_left hr (codingScale_pos k ε hδ).le |
| have hg := mul_le_mul_of_nonneg_left hh (by positivity : 0≤1+|9*growthBase k|) |
| dsimp only [codingRadius] |
| exact hg.trans (by simpa only [mul_one,one_mul,mul_comm] using codingScale_global k ε δ) |
| lemma codingRadius_le_one (k χ ε : ℝ) {δ : ℝ} (hδ : 0<δ) (z : Torus) : codingRadius k χ ε δ z≤1 := by |
| have hh := codingRadius_small k χ ε hδ z |
| have hp := codingRadius_pos k χ ε hδ z |
| nlinarith [abs_nonneg (9*growthBase k)] |
| lemma measurable_codingRadius (k χ ε δ : ℝ) : Measurable (codingRadius k χ ε δ) := |
| measurable_const.mul ((measurable_chartRadius k χ ε).pow_const 2) |
| noncomputable def codingLevel (k χ ε : ℝ) (δ : ℝ) (hδ : 0<δ) (z : Torus) : ℕ := |
| dyadicLevel (codingRadius k χ ε δ z) (codingRadius_pos k χ ε hδ z) |
| lemma measurable_codingLevel (k χ ε δ : ℝ) (hδ : 0<δ) : Measurable (codingLevel k χ ε δ hδ) := |
| measurable_dyadicLevel _ (codingRadius_pos k χ ε hδ) (measurable_codingRadius k χ ε δ) |
| lemma codingLevel_size (k χ ε δ : ℝ) (hδ : 0<δ) (z : Torus) : |
| 1/(2 : ℝ)^(codingLevel k χ ε δ hδ z)≤codingRadius k χ ε δ z := |
| inv_dyadicLevel_le _ _ |
| lemma codingRadius_ratio (k χ ε : ℝ) {δ : ℝ} (hδ : 0<δ) (z : Torus) (hz : FineRegular k χ ε z) : |
| codingRadius k χ ε δ z≤Real.exp (2*ε)*codingRadius k χ ε δ (standardMap k z) ∧ |
| codingRadius k χ ε δ (standardMap k z)≤Real.exp (2*ε)*codingRadius k χ ε δ z := by |
| have he : 0<Real.exp ε := Real.exp_pos ε |
| have hex : (Real.exp ε)^2=Real.exp (2*ε) := by rw [sq,←Real.exp_add]; congr 1; ring |
| have hh : chartRadius k χ ε z≤Real.exp ε*chartRadius k χ ε (standardMap k z) := by |
| have ht := mul_le_mul_of_nonneg_left hz.2.2.2.2.2.1 he.le |
| rw [←mul_assoc,←Real.exp_add,add_neg_cancel,Real.exp_zero,one_mul] at ht |
| exact ht |
| have hh' : chartRadius k χ ε (standardMap k z)≤Real.exp ε*chartRadius k χ ε z := by |
| have ht := mul_le_mul_of_nonneg_left hz.2.2.2.2.2.2 he.le |
| rw [←mul_assoc,←Real.exp_add,add_neg_cancel,Real.exp_zero,one_mul] at ht |
| exact ht |
| have general (x y : Torus) (h : chartRadius k χ ε x≤Real.exp ε*chartRadius k χ ε y) : |
| codingRadius k χ ε δ x≤Real.exp (2*ε)*codingRadius k χ ε δ y := by |
| have hs := sq_le_sq₀ (chartRadius_pos k χ ε x).le (mul_nonneg he.le (chartRadius_pos k χ ε y).le) |>.mpr h |
| rw [mul_pow,hex] at hs |
| have ht := mul_le_mul_of_nonneg_left hs (codingScale_pos k ε hδ).le |
| simpa only [codingRadius,mul_left_comm] using ht |
| exact ⟨general _ _ hh,general _ _ hh'⟩ |
| lemma codingLevel_step (k χ ε δ : ℝ) (hδ : 0<δ) (z : Torus) (hz : FineRegular k χ ε z) |
| (J : ℕ) (hJ : Real.exp (2*ε)≤(2 : ℝ)^J) : |
| codingLevel k χ ε δ hδ (standardMap k z)≤codingLevel k χ ε δ hδ z+J ∧ |
| codingLevel k χ ε δ hδ z≤codingLevel k χ ε δ hδ (standardMap k z)+J := by |
| have hr := codingRadius_ratio k χ ε hδ z hz |
| exact ⟨dyadicLevel_mono_bound _ _ J (hr.1.trans (mul_le_mul_of_nonneg_right hJ (codingRadius_pos k χ ε hδ _).le)), |
| dyadicLevel_mono_bound _ _ J (hr.2.trans (mul_le_mul_of_nonneg_right hJ (codingRadius_pos k χ ε hδ _).le))⟩ |
| lemma fineInverse_codingRadius (k χ ε : ℝ) {δ : ℝ} (hδ : 0<δ) (z : Torus) (hz : FineRegular k χ ε z) : |
| ‖fineInverse k χ ε δ z‖*codingRadius k χ ε δ z≤1/100 := by |
| let r := chartRadius k χ ε z |
| let η := fineScale k ε δ |
| let c := codingScale k ε δ |
| have hr : 0<r := chartRadius_pos k χ ε z |
| have hη : 0<η := fineScale_pos k ε hδ |
| have hc : 0<c := codingScale_pos k ε hδ |
| have hb : ‖fineInverse k χ ε δ z‖≤(η*r)⁻¹*‖lyapunovCoframe k χ z‖ := |
| flatInverse_norm_le (fineSize_pos k χ ε hδ z).le _ |
| calc |
| _ ≤ ((η*r)⁻¹*‖lyapunovCoframe k χ z‖)*(c*r^2) := mul_le_mul_of_nonneg_right hb (by positivity) |
| _ = (c/η)*(r*‖lyapunovCoframe k χ z‖) := by field_simp |
| _ ≤ (c/η)*1 := mul_le_mul_of_nonneg_left hz.2.2.2.2.1 (by positivity) |
| _ ≤ 1/100 := by |
| rw [mul_one] |
| exact (div_le_iff₀ hη).mpr (by simpa only [div_eq_mul_inv,mul_comm,one_mul,c,η] using codingScale_le k ε hδ) |
|
|
| end StandardMapEntropy |
|
|
| end |
| section |
| namespace StandardMapEntropy.NonlinearStable |
| open Set Filter MeasureTheory |
| open scoped Topology NNReal ENNReal |
| variable {ℓ δ : ℝ≥0} |
|
|
| lemma Recurrence.linearRate_pos (r : Recurrence ℓ δ) : 0<(ℓ : ℝ) := |
| (abs_pos.mpr (inv_ne_zero (r.b_ne 0))).trans_le (r.inverse_bound 0) |
| lemma Recurrence.unstable_size_lower (r : Recurrence ℓ δ) (n : ℕ) : 1≤(ℓ : ℝ)*|r.b n| := by |
| have hh := mul_le_mul_of_nonneg_right (r.inverse_bound n) (abs_nonneg (r.b n)) |
| simpa only [abs_inv,inv_mul_cancel₀ (abs_ne_zero.mpr (r.b_ne n))] using hh |
| lemma Recurrence.unstable_size_ge_one (r : Recurrence ℓ δ) (h : ℓ+δ<1) (n : ℕ) : 1≤|r.b n| := by |
| have hℓ : (ℓ : ℝ)≤1 := NNReal.coe_le_coe.mpr ((le_add_of_nonneg_right (show (0 : ℝ≥0)≤δ from bot_le)).trans h.le) |
| exact (r.unstable_size_lower n).trans ((mul_le_mul_of_nonneg_right hℓ (abs_nonneg _)).trans_eq (one_mul _)) |
| lemma Recurrence.unstable_expansion (r : Recurrence ℓ δ) (h : ℓ+δ<1) (n : ℕ) : |
| 1≤((ℓ+δ : ℝ≥0) : ℝ)*(|r.b n|-(δ : ℝ)) := by |
| have hq : ((ℓ+δ : ℝ≥0) : ℝ)≤1 := NNReal.coe_le_coe.mpr h.le |
| have hb : ((ℓ+δ : ℝ≥0) : ℝ)≤|r.b n| := hq.trans (r.unstable_size_ge_one h n) |
| have hd := mul_nonneg δ.coe_nonneg (sub_nonneg.mpr hb) |
| have hl := r.unstable_size_lower n |
| simp only [NNReal.coe_add] at hd ⊢ |
| nlinarith |
|
|
| lemma Recurrence.step_snd_lower (r : Recurrence ℓ δ) (n : ℕ) (v w : Plane) : |
| |r.b n| *|v.2-w.2|≤|(r.step n v).2-(r.step n w).2|+(δ : ℝ)*‖v-w‖ := by |
| have hr := (r.lipschitz n).dist_le_mul v w |
| rw [dist_eq_norm,dist_eq_norm] at hr |
| have hri := (norm_snd_le (r.remainder n v-r.remainder n w)).trans hr |
| have he : r.b n*(v.2-w.2)=((r.step n v).2-(r.step n w).2)- |
| ((r.remainder n v).2-(r.remainder n w).2) := by dsimp only [Recurrence.step]; ring |
| rw [←abs_mul,he] |
| exact (abs_sub _ _).trans (add_le_add le_rfl hri) |
|
|
| lemma Recurrence.unstable_cone_step (r : Recurrence ℓ δ) (h : ℓ+δ<1) (n : ℕ) (v w : Plane) |
| (hcone : |v.1-w.1|≤|v.2-w.2|) : |
| |(r.step n v).1-(r.step n w).1|≤|(r.step n v).2-(r.step n w).2| ∧ |
| |v.2-w.2|≤((ℓ+δ : ℝ≥0) : ℝ)*|(r.step n v).2-(r.step n w).2| := by |
| have hvw : ‖v-w‖=|v.2-w.2| := max_eq_right hcone |
| have hl := r.step_snd_lower n v w |
| rw [hvw] at hl |
| have hl' : (|r.b n|-(δ : ℝ))*|v.2-w.2|≤|(r.step n v).2-(r.step n w).2| := by nlinarith |
| have hb : |v.2-w.2|≤((ℓ+δ : ℝ≥0) : ℝ)*|(r.step n v).2-(r.step n w).2| := by |
| calc |
| _ ≤ (((ℓ+δ : ℝ≥0) : ℝ)*(|r.b n|-(δ : ℝ)))*|v.2-w.2| := by |
| simpa only [one_mul] using mul_le_mul_of_nonneg_right (r.unstable_expansion h n) (abs_nonneg (v.2-w.2)) |
| _ = ((ℓ+δ : ℝ≥0) : ℝ)*((|r.b n|-(δ : ℝ))*|v.2-w.2|) := mul_assoc _ _ _ |
| _ ≤ _ := mul_le_mul_of_nonneg_left hl' (ℓ+δ).coe_nonneg |
| refine ⟨?_,hb⟩ |
| have hf := step_fst_sub_bound r n v w |
| rw [hvw] at hf |
| have hq : ((ℓ+δ : ℝ≥0) : ℝ)≤1 := NNReal.coe_le_coe.mpr h.le |
| have hc : ((ℓ+δ : ℝ≥0) : ℝ)*|v.2-w.2|≤|v.2-w.2| := |
| (mul_le_mul_of_nonneg_right hq (abs_nonneg _)).trans_eq (one_mul _) |
| have hb' : ((ℓ+δ : ℝ≥0) : ℝ)*|(r.step n v).2-(r.step n w).2|≤|(r.step n v).2-(r.step n w).2| := |
| (mul_le_mul_of_nonneg_right hq (abs_nonneg _)).trans_eq (one_mul _) |
| exact hf.trans (hc.trans (hb.trans hb')) |
|
|
| noncomputable def Recurrence.orbit (r : Recurrence ℓ δ) (v : Plane) : ℕ → Plane |
| | 0 => v |
| | n+1 => r.step n (r.orbit v n) |
| lemma Recurrence.orbit_continuous (r : Recurrence ℓ δ) (n : ℕ) : Continuous (fun v => r.orbit v n) := by |
| induction n with |
| | zero => exact continuous_id |
| | succ n ih => |
| exact ((continuous_const.mul ih.fst).add ((r.lipschitz n).continuous.comp ih).fst).prodMk |
| ((continuous_const.mul ih.snd).add ((r.lipschitz n).continuous.comp ih).snd) |
|
|
| lemma Recurrence.orbit_unstable_cone (r : Recurrence ℓ δ) (h : ℓ+δ<1) (s t u : ℝ) (n : ℕ) : |
| |(r.orbit (s,t) n).1-(r.orbit (s,u) n).1|≤|(r.orbit (s,t) n).2-(r.orbit (s,u) n).2| ∧ |
| |t-u|≤((ℓ+δ : ℝ≥0) : ℝ)^n*|(r.orbit (s,t) n).2-(r.orbit (s,u) n).2| := by |
| induction n with |
| | zero => exact ⟨by simp only [Recurrence.orbit,sub_self,abs_zero]; positivity,by simp only [Recurrence.orbit,pow_zero,one_mul,le_refl]⟩ |
| | succ n ih => |
| have hh := r.unstable_cone_step h n (r.orbit (s,t) n) (r.orbit (s,u) n) ih.1 |
| refine ⟨hh.1,ih.2.trans ?_⟩ |
| exact (mul_le_mul_of_nonneg_left hh.2 (pow_nonneg (ℓ+δ).coe_nonneg n)).trans_eq (by |
| simp only [Recurrence.orbit,pow_succ]; ring) |
|
|
| def Recurrence.trappedBox (r : Recurrence ℓ δ) (n : ℕ) : Set Plane := |
| {v | ∀ j : ℕ, j≤n → ‖r.orbit v j‖≤1} |
| lemma Recurrence.trappedBox_closed (r : Recurrence ℓ δ) (n : ℕ) : IsClosed (r.trappedBox n) := by |
| have he : r.trappedBox n=⋂ j : ℕ, ⋂ (_ : j≤n), {v | ‖r.orbit v j‖≤1} := by ext v; simp only [Recurrence.trappedBox,mem_ofPred_eq,mem_iInter] |
| rw [he] |
| exact isClosed_iInter fun j => isClosed_iInter fun _ => isClosed_le (r.orbit_continuous j).norm continuous_const |
| lemma Recurrence.trappedBox_fiber_diameter (r : Recurrence ℓ δ) (h : ℓ+δ<1) (n : ℕ) |
| {s t u : ℝ} (ht : (s,t)∈r.trappedBox n) (hu : (s,u)∈r.trappedBox n) : |
| |t-u|≤2*((ℓ+δ : ℝ≥0) : ℝ)^n := by |
| have htn := (norm_snd_le (r.orbit (s,t) n)).trans (ht n le_rfl) |
| have hun := (norm_snd_le (r.orbit (s,u) n)).trans (hu n le_rfl) |
| simp only [Real.norm_eq_abs] at htn hun |
| have hb : |(r.orbit (s,t) n).2-(r.orbit (s,u) n).2|≤2 := (abs_sub _ _).trans (by linarith) |
| exact (r.orbit_unstable_cone h s t u n).2.trans |
| ((mul_le_mul_of_nonneg_left hb (pow_nonneg (ℓ+δ).coe_nonneg n)).trans_eq (by ring)) |
|
|
| end StandardMapEntropy.NonlinearStable |
|
|
| end |
| section |
| namespace StandardMapEntropy.NonlinearStable |
| open Set Filter MeasureTheory |
| open scoped Topology NNReal ENNReal |
| variable {ℓ δ : ℝ≥0} |
|
|
| lemma Recurrence.trappedBox_fiber_volume (r : Recurrence ℓ δ) (h : ℓ+δ<1) (n : ℕ) (s : ℝ) : |
| volume (Prod.mk s ⁻¹' r.trappedBox n)≤ENNReal.ofReal (2*((ℓ+δ : ℝ≥0) : ℝ)^n) := by |
| apply (Real.volume_le_diam _).trans |
| apply Metric.ediam_le_iff.mpr |
| intro t ht u hu |
| rw [edist_dist,Real.dist_eq] |
| exact ENNReal.ofReal_le_ofReal (r.trappedBox_fiber_diameter h n ht hu) |
|
|
| lemma Recurrence.trappedBox_fiber_empty (r : Recurrence ℓ δ) (n : ℕ) {s : ℝ} (hs : s∉Icc (-1 : ℝ) 1) : |
| Prod.mk s ⁻¹' r.trappedBox n=∅ := by |
| apply Set.eq_empty_iff_forall_notMem.mpr |
| intro t ht |
| have hh := (norm_fst_le (s,t)).trans (ht 0 (Nat.zero_le n)) |
| exact hs (abs_le.mp hh) |
|
|
| theorem Recurrence.trappedBox_volume (r : Recurrence ℓ δ) (h : ℓ+δ<1) (n : ℕ) : |
| volume (r.trappedBox n)≤ENNReal.ofReal (4*((ℓ+δ : ℝ≥0) : ℝ)^n) := by |
| change ((volume : Measure ℝ).prod volume) (r.trappedBox n)≤_ |
| rw [Measure.prod_apply (r.trappedBox_closed n).measurableSet] |
| calc |
| _ ≤ ∫⁻ s : ℝ, (Icc (-1 : ℝ) 1).indicator |
| (fun _ => ENNReal.ofReal (2*((ℓ+δ : ℝ≥0) : ℝ)^n)) s ∂volume := by |
| apply lintegral_mono |
| intro s |
| change volume (Prod.mk s ⁻¹' r.trappedBox n)≤_ |
| by_cases hs : s∈Icc (-1 : ℝ) 1 |
| · rw [indicator_of_mem hs] |
| exact r.trappedBox_fiber_volume h n s |
| · rw [indicator_of_notMem hs,r.trappedBox_fiber_empty n hs,measure_empty] |
| _ = _ := by |
| rw [lintegral_indicator measurableSet_Icc,setLIntegral_const,Real.volume_Icc, |
| ←ENNReal.ofReal_mul (by positivity)] |
| congr 1 |
| ring |
|
|
| end StandardMapEntropy.NonlinearStable |
|
|
| end |
| section |
| namespace StandardMapEntropy |
| open MeasureTheory Set Filter |
| open scoped Topology NNReal ENNReal |
|
|
| lemma fine_nonlinearity_coefficient (k χ ε : ℝ) {δ : ℝ} (hδ : 0<δ) (z : Torus) |
| (hz : FineRegular k χ ε z) (hn : FineRegular k χ ε (standardMap k z)) : |
| ‖fineInverse k χ ε δ (standardMap k z)‖*derivativeVariationBound k*‖fineFrame k χ ε δ z‖^2≤δ := by |
| let a := chartRadius k χ ε z |
| let b := chartRadius k χ ε (standardMap k z) |
| let c := ‖lyapunovCoframe k χ (standardMap k z)‖ |
| let η := fineScale k ε δ |
| have ha : 0<a := chartRadius_pos k χ ε z |
| have hb : 0<b := chartRadius_pos k χ ε (standardMap k z) |
| have hη : 0<η := fineScale_pos k ε hδ |
| have hc : 0≤c := norm_nonneg _ |
| have hbc : b*c≤1 := hn.2.2.2.2.1 |
| have hab : a/b≤Real.exp ε := slow_radius_ratio ha hb hz.2.2.2.2.2.1 |
| have hD := derivativeVariationBound_nonneg k |
| have hF : ‖fineFrame k χ ε δ z‖≤2*(η*a) := fineFrame_norm_le k χ ε hδ z |
| have hC : ‖fineInverse k χ ε δ (standardMap k z)‖≤(η*b)⁻¹*c := |
| flatInverse_norm_le (fineSize_pos k χ ε hδ _).le _ |
| have hEF : (Real.exp ε)^2=Real.exp (2*ε) := by rw [pow_two,←Real.exp_add]; congr 1; ring |
| have hsqr : (a/b)^2≤Real.exp (2*ε) := by |
| rw [←hEF] |
| exact pow_le_pow_left₀ (div_nonneg ha.le hb.le) hab 2 |
| calc |
| _ ≤ ((η*b)⁻¹*c)*derivativeVariationBound k*(2*(η*a))^2 := |
| mul_le_mul (mul_le_mul_of_nonneg_right hC hD) |
| (pow_le_pow_left₀ (norm_nonneg _) hF 2) (sq_nonneg _) |
| (mul_nonneg (mul_nonneg (inv_nonneg.mpr (mul_nonneg hη.le hb.le)) hc) hD) |
| _ = (4*derivativeVariationBound k*η)*((b*c)*(a/b)^2) := by |
| field_simp [ne_of_gt hη,ne_of_gt hb]; ring |
| _ ≤ (4*derivativeVariationBound k*η)*(1*Real.exp (2*ε)) := |
| mul_le_mul_of_nonneg_left (mul_le_mul hbc hsqr (sq_nonneg _) (by norm_num)) (by positivity) |
| _ = δ*(derivativeVariationBound k/(2*(1+derivativeVariationBound k))) := by |
| dsimp [η,fineScale] |
| field_simp [ne_of_gt (Real.exp_pos (2*ε)),ne_of_gt (show 0<1+derivativeVariationBound k by linarith)] |
| ring |
| _ ≤ δ*1 := by |
| apply mul_le_mul_of_nonneg_left _ hδ.le |
| apply (div_le_iff₀ (show 0<2*(1+derivativeVariationBound k) by linarith)).mpr |
| linarith |
| _ = _ := mul_one _ |
|
|
| noncomputable def fineRemainder (k χ ε δ : ℝ) (z : ℂ) : RealPlane → RealPlane := |
| flatRemainder k z (fineFrame k χ ε δ (complexProjection z)) |
| (fineInverse k χ ε δ (standardMap k (complexProjection z))) |
| @[simp] lemma fineRemainder_zero (k χ ε δ : ℝ) (z : ℂ) : fineRemainder k χ ε δ z 0=0 := |
| flatRemainder_zero _ _ _ _ |
| lemma fineRemainder_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} (hv : ‖v‖≤1) (hw : ‖w‖≤1) : |
| ‖fineRemainder k χ ε δ z v-fineRemainder k χ ε δ z w‖≤δ*‖v-w‖ := |
| (flatRemainder_sub_bound k z _ _ hv hw).trans |
| (mul_le_mul_of_nonneg_right (fine_nonlinearity_coefficient k χ ε hδ _ hz hn) (norm_nonneg _)) |
| lemma fine_derivative_diagonal (k χ ε δ : ℝ) (z : Torus) (hz : FineRegular k χ ε z) (v : RealPlane) : |
| fineInverse k χ ε δ (standardMap k z) (standardDerivative k z (fineFrame k χ ε δ z v)) = |
| (fineStableMultiplier k χ ε δ z*v.1,fineUnstableMultiplier k χ ε δ z*v.2) := |
| flat_diagonal _ _ _ hz.2.1 v |
| lemma fine_step (k χ ε δ : ℝ) (z : ℂ) (hz : FineRegular k χ ε (complexProjection z)) (v : RealPlane) : |
| fineInverse k χ ε δ (standardMap k (complexProjection z)) |
| (standardLift k (z+fineFrame k χ ε δ (complexProjection z) v)-standardLift k z) = |
| (fineStableMultiplier k χ ε δ (complexProjection z)*v.1+(fineRemainder k χ ε δ z v).1, |
| fineUnstableMultiplier k χ ε δ (complexProjection z)*v.2+(fineRemainder k χ ε δ z v).2) := by |
| have hdiff : standardLift k (z+fineFrame k χ ε δ (complexProjection z) v)-standardLift k z = |
| standardDerivative k (complexProjection z) (fineFrame k χ ε δ (complexProjection z) v)+ |
| liftRemainder k z (fineFrame k χ ε δ (complexProjection z) v) := by unfold liftRemainder; abel |
| rw [hdiff,map_add,fine_derivative_diagonal k χ ε δ _ hz] |
| rfl |
| lemma fine_lift_step (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) : |
| standardLift k (z+fineFrame k χ ε δ (complexProjection z) v) = standardLift k z+ |
| fineFrame k χ ε δ (standardMap k (complexProjection z)) |
| (fineStableMultiplier k χ ε δ (complexProjection z)*v.1+(fineRemainder k χ ε δ z v).1, |
| fineUnstableMultiplier k χ ε δ (complexProjection z)*v.2+(fineRemainder k χ ε δ z v).2) := by |
| have hh := congrArg (fineFrame k χ ε δ (standardMap k (complexProjection z))) (fine_step k χ ε δ z hz v) |
| rw [fineFrame_inverse k χ ε hδ _ hn.1] at hh |
| exact (sub_eq_iff_eq_add.mp hh).trans (add_comm _ _) |
|
|
| end StandardMapEntropy |
|
|
| end |
| section |
| namespace StandardMapEntropy |
| open MeasureTheory Set Filter |
| open scoped Topology NNReal ENNReal |
| open NonlinearStable |
|
|
| noncomputable def fineLinearRate (χ ε : ℝ) : ℝ≥0 := ⟨Real.exp (-χ+ε),(Real.exp_pos _).le⟩ |
| noncomputable def fineErrorRate (δ : ℝ) (hδ : 0<δ) : ℝ≥0 := ⟨δ,hδ.le⟩ |
| lemma fine_regular_lift_next (k χ ε : ℝ) (w : ℂ) |
| (hw : ∀ n : ℕ, FineRegular k χ ε (complexProjection ((standardLift k)^[n] w))) (n : ℕ) : |
| FineRegular k χ ε (standardMap k (complexProjection ((standardLift k)^[n] w))) := by |
| simpa only [Function.iterate_succ_apply',complexProjection_standardLift] using hw (n+1) |
| noncomputable def fineLocalRecurrence (k χ ε δ : ℝ) (hδ : 0<δ) (w : ℂ) |
| (hw : ∀ n : ℕ, FineRegular k χ ε (complexProjection ((standardLift k)^[n] w))) : |
| LocalRecurrence (fineLinearRate χ ε) (fineErrorRate δ hδ) where |
| a n := fineStableMultiplier k χ ε δ (complexProjection ((standardLift k)^[n] w)) |
| b n := fineUnstableMultiplier k χ ε δ (complexProjection ((standardLift k)^[n] w)) |
| remainder n := fineRemainder k χ ε δ ((standardLift k)^[n] w) |
| stable_bound n := fineStableMultiplier_bound k χ ε hδ _ (hw n) |
| inverse_bound n := fineUnstableMultiplier_inverse_bound k χ ε hδ _ (hw n) |
| b_ne n := fineUnstableMultiplier_ne k χ ε hδ _ (hw n) |
| local_lipschitz n _ _ hv hu := fineRemainder_lipschitz k χ ε hδ _ (hw n) (fine_regular_lift_next k χ ε w hw n) hv hu |
| zero _ := fineRemainder_zero _ _ _ _ _ |
|
|
| noncomputable def fineCoordinate (k χ ε δ : ℝ) (w p : ℂ) (n : ℕ) : RealPlane := |
| fineInverse k χ ε δ (complexProjection ((standardLift k)^[n] w)) |
| ((standardLift k)^[n] p-(standardLift k)^[n] w) |
| lemma fineCoordinate_initial (k χ ε : ℝ) {δ : ℝ} (hδ : 0<δ) (w : ℂ) |
| (hw : FineRegular k χ ε (complexProjection w)) (v : RealPlane) : |
| fineCoordinate k χ ε δ w (w+fineFrame k χ ε δ (complexProjection w) v) 0=v := by |
| change fineInverse k χ ε δ (complexProjection w) (w+fineFrame k χ ε δ (complexProjection w) v-w)=v |
| rw [add_sub_cancel_left,fineInverse_frame k χ ε hδ _ hw.1] |
| lemma fineCoordinate_step (k χ ε δ : ℝ) (hδ : 0<δ) (w p : ℂ) |
| (hw : ∀ n : ℕ, FineRegular k χ ε (complexProjection ((standardLift k)^[n] w))) (n : ℕ) : |
| fineCoordinate k χ ε δ w p (n+1)= |
| (fineLocalRecurrence k χ ε δ hδ w hw).step n (fineCoordinate k χ ε δ w p n) := by |
| have he : (standardLift k)^[n] p=(standardLift k)^[n] w+ |
| fineFrame k χ ε δ (complexProjection ((standardLift k)^[n] w)) (fineCoordinate k χ ε δ w p n) := by |
| dsimp only [fineCoordinate] |
| rw [fineFrame_inverse k χ ε hδ _ (hw n).1] |
| abel |
| conv_lhs => simp only [fineCoordinate,Function.iterate_succ_apply'] |
| rw [he] |
| simpa only [Function.iterate_succ_apply',complexProjection_standardLift,fineLocalRecurrence,LocalRecurrence.step] using |
| fine_step k χ ε δ ((standardLift k)^[n] w) (hw n) (fineCoordinate k χ ε δ w p n) |
|
|
| def fineTrapped (k χ ε δ : ℝ) (w : ℂ) (n : ℕ) : Set RealPlane := |
| {v | ∀ j : ℕ, j≤n → ‖fineCoordinate k χ ε δ w |
| (w+fineFrame k χ ε δ (complexProjection w) v) j‖≤1} |
| lemma fineTrapped_subset (k χ ε δ : ℝ) (hδ : 0<δ) (w : ℂ) |
| (hw : ∀ n : ℕ, FineRegular k χ ε (complexProjection ((standardLift k)^[n] w))) (n : ℕ) : |
| fineTrapped k χ ε δ w n ⊆ (fineLocalRecurrence k χ ε δ hδ w hw).extend.trappedBox n := by |
| intro v hv |
| have he : ∀ j : ℕ, 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 |
| intro j |
| induction j with |
| | zero => intro _; exact (fineCoordinate_initial k χ ε hδ w (hw 0) v).symm |
| | succ j ih => |
| intro hj |
| rw [Recurrence.orbit,ih (by omega),extend_step_eq _ j (hv j (by omega)),fineCoordinate_step] |
| intro j hj |
| rw [he j hj] |
| exact hv j hj |
|
|
| theorem fineTrapped_volume (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))) (n : ℕ) : |
| volume (fineTrapped k χ ε δ w n)≤ENNReal.ofReal (4*(Real.exp (-χ+ε)+δ)^n) := by |
| have hq' : fineLinearRate χ ε+fineErrorRate δ hδ<1 := hq |
| exact (measure_mono (fineTrapped_subset k χ ε δ hδ w hw n)).trans |
| ((fineLocalRecurrence k χ ε δ hδ w hw).extend.trappedBox_volume hq' n) |
|
|
| lemma fine_parameters_for_rate {a χ : ℝ} (ha : 0<a) (hχ : a<χ) : |
| ∃ ε δ : ℝ, 0<ε ∧ 0<δ ∧ Real.exp (-χ+ε)+δ<1 ∧ Real.exp (-χ+ε)+δ≤Real.exp (-a) := by |
| let ε := (χ-a)/2 |
| have he : 0<ε := by dsimp [ε]; linarith |
| have hx : Real.exp (-χ+ε)<Real.exp (-a) := Real.exp_lt_exp.mpr (by dsimp [ε]; linarith) |
| have h1 : Real.exp (-a)<1 := Real.exp_lt_one_iff.mpr (neg_neg_of_pos ha) |
| refine ⟨ε,(Real.exp (-a)-Real.exp (-χ+ε))/2,he,?_,?_,?_⟩ <;> linarith |
|
|
| theorem ae_sharp_fine_volume (k : ℝ) (hk : 0≤k) {a χ : ℝ} (ha : 0<a) (hχ : a<χ) : |
| ∃ ε δ : ℝ, 0<ε ∧ 0<δ ∧ |
| ∀ᵐ z ∂area.restrict (spectralGapRegion k hk χ), ∀ w : ℂ, complexProjection w=z → ∀ n : ℕ, |
| volume (fineTrapped k χ ε δ w n)≤ENNReal.ofReal (4*Real.exp (-a*(n : ℝ))) := by |
| obtain ⟨ε,δ,hε,hδ,hq,hqa⟩ := fine_parameters_for_rate ha hχ |
| refine ⟨ε,δ,hε,hδ,?_⟩ |
| filter_upwards [ae_fineRegular_all_iterates k hk (ha.trans hχ) hε] with z hz |
| intro w hw n |
| have hr (j : ℕ) : FineRegular k χ ε (complexProjection ((standardLift k)^[j] w)) := by |
| simpa only [complexProjection_iterate,hw] using (hz j).1 |
| apply (fineTrapped_volume k χ ε δ hδ hq w hr n).trans |
| apply ENNReal.ofReal_le_ofReal |
| apply mul_le_mul_of_nonneg_left _ (by norm_num) |
| calc |
| _ ≤ (Real.exp (-a))^n := pow_le_pow_left₀ (by positivity) hqa n |
| _ = _ := by rw [←Real.exp_nat_mul]; congr 1; ring |
|
|
| end StandardMapEntropy |
|
|
| end |
| end |
|
|
| end OAI |
|
|
