| import OAI.Dynamics.StandardMap.EntropyEndpoint |
| import OAI.Dynamics.StandardMap.Holonomy.FineStableGeometry |
| import OAI.Dynamics.StandardMap.Holonomy.FiniteOrbitDecay |
|
|
| namespace OAI |
|
|
| section |
| section |
| namespace StandardMapEntropy.NonlinearStable |
| open Set Filter MeasureTheory |
| open scoped Topology NNReal ENNReal |
| variable {ℓ δ : ℝ≥0} |
|
|
| lemma LocalRecurrence.trapped_affine_derivative (r : LocalRecurrence ℓ δ) (h : ℓ+δ<1) |
| (hδ : (δ : ℝ)≤1/2) (D : ℕ → Plane → Plane →L[ℝ] Plane) |
| (hd : ∀ n v, HasFDerivAt (r.remainder n) (D n v) v) |
| (hsmall : ∀ n v, ‖v‖≤1 → ‖D n v‖≤(δ : ℝ)*‖v‖) |
| (p V : Plane) (hcone : |V.1|≤|V.2|) (hV0 : V.2≠0) |
| (x : ℝ) {N : ℕ} (hv : p+x•V∈r.extend.trappedBox N) : |
| ∃ J : ℝ, HasDerivAt (fun t : ℝ => (r.trueOrbit (p+t•V) N).2) J x ∧ |
| J≠0 ∧ |Real.log |J|-Real.log |V.2|-(∑ j ∈ Finset.range N,Real.log |r.b j|)|≤ |
| 4*(δ : ℝ)/(1-((ℓ+δ : ℝ≥0) : ℝ)) := by |
| classical |
| let L (n : ℕ) : Plane →L[ℝ] Plane := if n<N then D n (r.trueOrbit (p+x•V) n) else 0 |
| have hLs (n : ℕ) (hn : n<N) : ‖L n‖≤(δ : ℝ)*‖r.extend.orbit (p+x•V) n‖ := by |
| dsimp only [L] |
| rw [ite_eq_left hn,r.trueOrbit_eq_extend hv hn.le] |
| exact hsmall n _ (hv n hn.le) |
| have hL (n : ℕ) : ‖L n‖≤δ := by |
| by_cases hn : n<N |
| · exact (hLs n hn).trans ((mul_le_mul_of_nonneg_left (hv n hn.le) δ.coe_nonneg).trans_eq (mul_one _)) |
| · simp only [L,ite_eq_right hn,norm_zero]; exact δ.coe_nonneg |
| let R := r.linearized L hL |
| refine ⟨(R.orbit V N).2,?_,R.orbit_snd_ne_zero h hcone hV0 N, |
| r.linearized_cone_log_distortion h hδ L hL hv hcone hV0 hLs⟩ |
| exact (r.trueOrbit_affine_hasDerivAt D hd L hL p V x (fun n hn => ite_eq_left hn) (le_refl N)).snd |
|
|
| theorem LocalRecurrence.affine_interval_distortion (r : LocalRecurrence ℓ δ) (h : ℓ+δ<1) |
| (hδ : (δ : ℝ)≤1/2) (D : ℕ → Plane → Plane →L[ℝ] Plane) |
| (hd : ∀ n v, HasFDerivAt (r.remainder n) (D n v) v) |
| (hsmall : ∀ n v, ‖v‖≤1 → ‖D n v‖≤(δ : ℝ)*‖v‖) |
| (p V : Plane) (hcone : |V.1|≤|V.2|) (hV0 : V.2≠0) |
| {t u : ℝ} (htu : t≤u) {N : ℕ} |
| (ht : ∀ n : ℕ,n≤N → ‖r.extend.orbit (p+t•V) n‖≤1/3) |
| (hu : ∀ n : ℕ,n≤N → ‖r.extend.orbit (p+u•V) n‖≤1/3) : |
| let A := Real.log |V.2|+∑ j ∈ Finset.range N,Real.log |r.b j| |
| let C := 4*(δ : ℝ)/(1-((ℓ+δ : ℝ≥0) : ℝ)) |
| Real.exp (A-C)*(u-t)≤|(r.trueOrbit (p+u•V) N).2-(r.trueOrbit (p+t•V) N).2| ∧ |
| |(r.trueOrbit (p+u•V) N).2-(r.trueOrbit (p+t•V) N).2|≤Real.exp (A+C)*(u-t) := by |
| dsimp only |
| rcases htu.eq_or_lt with he | htu |
| · subst u; simp |
| let f : ℝ → ℝ := fun v => (r.trueOrbit (p+v•V) N).2 |
| have hfc : Continuous f := ((r.trueOrbit_continuous (fun n => continuous_iff_continuousAt.mpr |
| fun v => (hd n v).continuousAt) N).comp (continuous_const.add (continuous_id.smul continuous_const))).snd |
| have htrap (v : ℝ) (hv : v∈Icc t u) : (p+v•V)∈r.extend.trappedBox N := |
| r.extend.affine_interval_trapped h hcone hV0 p hv.1 hv.2 ht hu |
| have hf (v : ℝ) (hv : v∈Icc t u) : ∃ J : ℝ, HasDerivAt f J v ∧ J≠0 ∧ |
| |Real.log |J|-Real.log |V.2|-(∑ j ∈ Finset.range N,Real.log |r.b j|)|≤ |
| 4*(δ : ℝ)/(1-((ℓ+δ : ℝ≥0) : ℝ)) := |
| r.trapped_affine_derivative h hδ D hd hsmall p V hcone hV0 v (htrap v hv) |
| obtain ⟨c,hc,he⟩ := exists_deriv_eq_slope f htu hfc.continuousOn |
| (fun v hv => (hf v ⟨hv.1.le,hv.2.le⟩).choose_spec.1.differentiableAt.differentiableWithinAt) |
| obtain ⟨J,hJ,hJn,hJL⟩ := hf c ⟨hc.1.le,hc.2.le⟩ |
| rw [hJ.deriv] at he |
| have hab : f u-f t=J*(u-t) := (eq_div_iff (sub_ne_zero.mpr htu.ne')).mp he |>.symm |
| have heq : |f u-f t|=|J| *(u-t) := by rw [hab,abs_mul,abs_of_pos (sub_pos.mpr htu)] |
| change _≤|f u-f t| ∧ |f u-f t|≤_ |
| rw [heq] |
| have hJL' : |Real.log |J|-(Real.log |V.2|+∑ j ∈ Finset.range N,Real.log |r.b j|)|≤ |
| 4*(δ : ℝ)/(1-((ℓ+δ : ℝ≥0) : ℝ)) := by |
| convert hJL using 1 |
| congr 1 |
| ring |
| have hb := exp_bounds_of_log_error hJn hJL' |
| exact ⟨mul_le_mul_of_nonneg_right hb.1 (sub_nonneg.mpr htu.le), |
| mul_le_mul_of_nonneg_right hb.2 (sub_nonneg.mpr htu.le)⟩ |
|
|
| end StandardMapEntropy.NonlinearStable |
|
|
| end |
| section |
| namespace StandardMapEntropy.NonlinearStable |
| open Set Filter MeasureTheory |
| open scoped Topology NNReal ENNReal |
| variable {ℓ δ : ℝ≥0} |
|
|
| lemma Recurrence.pair_norm_eq_snd (r : Recurrence ℓ δ) (h : ℓ+δ<1) {v w : Plane} |
| (hv : |v.1-w.1|≤|v.2-w.2|) (n : ℕ) : |
| ‖r.orbit v n-r.orbit w n‖=|(r.orbit v n).2-(r.orbit w n).2| := by |
| change max _ _ = _ |
| exact max_eq_right (r.orbit_pair_cone h hv n).1 |
|
|
| lemma Recurrence.exists_orbit_escape (r : Recurrence ℓ δ) (h : ℓ+δ<1) {v w : Plane} |
| (hv : |v.1-w.1|≤|v.2-w.2|) (hne : v.2≠w.2) |
| {R : ℝ} (hw : ∀ n, ‖r.orbit w n‖≤R) : ∃ n, R<‖r.orbit v n‖ := by |
| by_contra hn |
| push Not at hn |
| have hbound (n : ℕ) : |v.2-w.2|≤((ℓ+δ : ℝ≥0) : ℝ)^n*(2*R) := by |
| have hb : |(r.orbit v n).2-(r.orbit w n).2|≤2*R := |
| (abs_sub _ _).trans (by |
| have h₁ := (norm_snd_le (r.orbit v n)).trans (hn n) |
| have h₂ := (norm_snd_le (r.orbit w n)).trans (hw n) |
| simp only [Real.norm_eq_abs] at h₁ h₂ |
| linarith) |
| exact (r.orbit_pair_cone h hv n).2.trans (mul_le_mul_of_nonneg_left hb (by positivity)) |
| have ht := (tendsto_pow_atTop_nhds_zero_of_lt_one (ℓ+δ).coe_nonneg |
| (show ((ℓ+δ : ℝ≥0) : ℝ)<1 from h)).mul_const (2*R) |
| have hz : |v.2-w.2|≤0 := by simpa only [zero_mul] using |
| (le_of_tendsto_of_tendsto tendsto_const_nhds ht (Filter.Eventually.of_forall hbound)) |
| exact (abs_pos.mpr (sub_ne_zero.mpr hne)).not_ge hz |
|
|
| lemma last_trapped_time {P : ℕ → Prop} [DecidablePred P] (hbad : ∃ n, ¬P n) |
| {N₀ : ℕ} (hprefix : ∀ n, n≤N₀ → P n) : |
| ∃ N : ℕ, N₀≤N ∧ (∀ n, n≤N → P n) ∧ ¬P (N+1) := by |
| let j := Nat.find hbad |
| have hj : ¬P j := Nat.find_spec hbad |
| have hmin (n : ℕ) (hn : n<j) : P n := by |
| exact not_not.mp (Nat.find_min hbad hn) |
| have hNj : N₀<j := by |
| by_contra hh |
| exact hj (hprefix j (le_of_not_gt hh)) |
| refine ⟨j-1,by omega,?_,?_⟩ |
| · intro n hn; exact hmin n (by omega) |
| · have he : j-1+1=j := by omega |
| rwa [he] |
|
|
| lemma Recurrence.escape_vertical_lower (r : Recurrence ℓ δ) (h : ℓ+δ<1) |
| {v w : Plane} (hc : |v.1-w.1|≤|v.2-w.2|) |
| {M a : ℝ} (hM : 0<M) (ha : 0<a) |
| (hstep : ∀ n u, ‖r.step n u‖≤M*a^n*‖u‖) {N : ℕ} |
| (hescape : 1/3<‖r.orbit v (N+1)‖) |
| (hw : ‖r.orbit w N‖≤1/(6*M*a^N)) : |
| 1/(6*M*a^N)< |(r.orbit v N).2-(r.orbit w N).2| := by |
| have hp : 0<M*a^N := mul_pos hM (pow_pos ha N) |
| have hl : 1/(3*M*a^N)<‖r.orbit v N‖ := by |
| have hh := hescape.trans_le (hstep N (r.orbit v N)) |
| apply (div_lt_iff₀ (by positivity : 0<3*M*a^N)).mpr |
| nlinarith |
| have hh := norm_add_le (r.orbit v N-r.orbit w N) (r.orbit w N) |
| rw [sub_add_cancel,r.pair_norm_eq_snd h hc] at hh |
| have he : 1/(3*M*a^N)=1/(6*M*a^N)+1/(6*M*a^N) := by field_simp; ring |
| rw [he] at hl |
| linarith |
|
|
| lemma terminal_size_comparison {p₀ p₁ q₀ q₁ ρ : ℝ} (hρ : 0<ρ) |
| (hl : ρ< |(p₁-p₀)| ∨ ρ< |(q₁-q₀)|) |
| (hpair₀ : |p₀-q₀|≤ρ/4) (hpair₁ : |p₁-q₁|≤ρ/4) : |
| |q₁-q₀|≤2* |p₁-p₀| ∧ |p₁-p₀|≤2* |q₁-q₀| := by |
| have he : (p₁-p₀)-(q₁-q₀)=(p₁-q₁)-(p₀-q₀) := by ring |
| have hh : abs (abs (p₁-p₀)-abs (q₁-q₀))≤ρ/2 := |
| (abs_abs_sub_abs_le_abs_sub _ _).trans (by |
| rw [he] |
| exact (abs_sub _ _).trans (by linarith)) |
| have hb := abs_le.mp hh |
| rcases hl with hl|hl <;> constructor <;> linarith |
|
|
| end StandardMapEntropy.NonlinearStable |
|
|
| end |
| section |
| namespace StandardMapEntropy.NonlinearStable |
| open Set Filter MeasureTheory |
| open scoped Topology NNReal ENNReal |
| variable {ℓ δ : ℝ≥0} |
|
|
| lemma LocalRecurrence.affine_interval_abs_distortion (r : LocalRecurrence ℓ δ) (h : ℓ+δ<1) |
| (hδ : (δ : ℝ)≤1/2) (D : ℕ → Plane → Plane →L[ℝ] Plane) |
| (hd : ∀ n v, HasFDerivAt (r.remainder n) (D n v) v) |
| (hsmall : ∀ n v, ‖v‖≤1 → ‖D n v‖≤(δ : ℝ)*‖v‖) |
| (p V : Plane) (hc : |V.1|≤|V.2|) (hV : V.2≠0) |
| {t u : ℝ} {N : ℕ} |
| (ht : ∀ n : ℕ,n≤N → ‖r.extend.orbit (p+t•V) n‖≤1/3) |
| (hu : ∀ n : ℕ,n≤N → ‖r.extend.orbit (p+u•V) n‖≤1/3) : |
| let A := Real.log |V.2|+∑ j ∈ Finset.range N,Real.log |r.b j| |
| let C := 4*(δ : ℝ)/(1-((ℓ+δ : ℝ≥0) : ℝ)) |
| Real.exp (A-C)* |u-t|≤|(r.extend.orbit (p+u•V) N).2-(r.extend.orbit (p+t•V) N).2| ∧ |
| |(r.extend.orbit (p+u•V) N).2-(r.extend.orbit (p+t•V) N).2|≤Real.exp (A+C)* |u-t| := by |
| dsimp only |
| have ht' : p+t•V∈r.extend.trappedBox N := fun n hn => (ht n hn).trans (by norm_num) |
| have hu' : p+u•V∈r.extend.trappedBox N := fun n hn => (hu n hn).trans (by norm_num) |
| rcases le_total t u with htu|hut |
| · have hh := r.affine_interval_distortion h hδ D hd hsmall p V hc hV htu ht hu |
| rw [r.trueOrbit_eq_extend ht' le_rfl,r.trueOrbit_eq_extend hu' le_rfl] at hh |
| simpa only [abs_of_nonneg (sub_nonneg.mpr htu)] using hh |
| · have hh := r.affine_interval_distortion h hδ D hd hsmall p V hc hV hut hu ht |
| rw [r.trueOrbit_eq_extend ht' le_rfl,r.trueOrbit_eq_extend hu' le_rfl] at hh |
| rw [abs_sub_comm u t,abs_sub_comm ((r.extend.orbit (p+u•V) N).2)] |
| simpa only [abs_of_nonneg (sub_nonneg.mpr hut)] using hh |
|
|
| lemma relative_interval_ratio {A C p q P Q : ℝ} |
| (hp : P≤Real.exp (A+C)*p) (hq : Real.exp (A-C)*q≤Q) (h : Q≤2*P) : |
| q≤2*Real.exp (2*C)*p := by |
| have he : Real.exp (A+C)=Real.exp (A-C)*Real.exp (2*C) := by |
| rw [←Real.exp_add]; congr 1; ring |
| rw [he] at hp |
| have hh : Real.exp (A-C)*q≤Real.exp (A-C)*(2*Real.exp (2*C)*p) := by nlinarith |
| exact (mul_le_mul_iff_right₀ (Real.exp_pos (A-C))).mp hh |
|
|
| theorem LocalRecurrence.finite_holonomy_ratio (r : LocalRecurrence ℓ δ) (h : ℓ+δ<1) |
| (hδ : (δ : ℝ)≤1/2) (D : ℕ → Plane → Plane →L[ℝ] Plane) |
| (hd : ∀ n v, HasFDerivAt (r.remainder n) (D n v) v) |
| (hsmall : ∀ n v, ‖v‖≤1 → ‖D n v‖≤(δ : ℝ)*‖v‖) |
| (p q V : Plane) (hc : |V.1|≤|V.2|) (hV : V.2≠0) |
| {t₀ t₁ u₀ u₁ : ℝ} {N : ℕ} |
| (hp₀ : ∀ n : ℕ,n≤N → ‖r.extend.orbit (p+t₀•V) n‖≤1/3) |
| (hp₁ : ∀ n : ℕ,n≤N → ‖r.extend.orbit (p+t₁•V) n‖≤1/3) |
| (hq₀ : ∀ n : ℕ,n≤N → ‖r.extend.orbit (q+u₀•V) n‖≤1/3) |
| (hq₁ : ∀ n : ℕ,n≤N → ‖r.extend.orbit (q+u₁•V) n‖≤1/3) |
| (hterminal : |(r.extend.orbit (q+u₁•V) N).2-(r.extend.orbit (q+u₀•V) N).2|≤ |
| 2* |(r.extend.orbit (p+t₁•V) N).2-(r.extend.orbit (p+t₀•V) N).2|) : |
| |u₁-u₀|≤(2*Real.exp (8*(δ : ℝ)/(1-((ℓ+δ : ℝ≥0) : ℝ))))* |t₁-t₀| := by |
| have hp := r.affine_interval_abs_distortion h hδ D hd hsmall p V hc hV hp₀ hp₁ |
| have hq := r.affine_interval_abs_distortion h hδ D hd hsmall q V hc hV hq₀ hq₁ |
| have hh := relative_interval_ratio hp.2 hq.1 hterminal |
| convert hh using 1 |
| congr 2 |
| ring_nf |
|
|
| end StandardMapEntropy.NonlinearStable |
|
|
| end |
| section |
| namespace StandardMapEntropy.NonlinearStable |
| open Set Filter MeasureTheory |
| open scoped Topology NNReal ENNReal |
| variable {ℓ δ : ℝ≥0} |
|
|
| lemma eventually_scaled_power_small {a b C d : ℝ} (ha : 0<a) (hb : 0≤b) |
| (hab : a*b<1) (hd : 0<d) : |
| ∀ᶠ n : ℕ in atTop, C*b^n≤d/a^n := by |
| have ht : Tendsto (fun n : ℕ => C*(a*b)^n) atTop (𝓝 0) := by |
| simpa using (tendsto_pow_atTop_nhds_zero_of_lt_one (mul_nonneg ha.le hb) hab).const_mul C |
| filter_upwards [ht.eventually (gt_mem_nhds hd)] with n hn |
| apply (le_div_iff₀ (pow_pos ha n)).mpr |
| rw [mul_pow] at hn |
| nlinarith |
|
|
| theorem LocalRecurrence.holonomy_local_bound (r : LocalRecurrence ℓ δ) (h : ℓ+δ<1) |
| (hδ : (δ : ℝ)≤1/2) (D : ℕ → Plane → Plane →L[ℝ] Plane) |
| (hd : ∀ n v, HasFDerivAt (r.remainder n) (D n v) v) |
| (hsmall : ∀ n v, ‖v‖≤1 → ‖D n v‖≤(δ : ℝ)*‖v‖) |
| (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) |
| {M a b B ρ : ℝ} (hM : 0<M) (ha : 0<a) (hb : 0≤b) (hb1 : b≤1) |
| (hρ : 0≤ρ) (hab : a*b<1) (haρ : a*ρ<1) |
| (hstep : ∀ n u, ‖r.extend.step n u‖≤M*a^n*‖u‖) |
| (hp₀ : ∀ n, ‖r.extend.orbit (p+t₀•V) n‖≤b^n/12) |
| (hq₀ : ∀ n, ‖r.extend.orbit (q+H t₀•V) n‖≤b^n/12) |
| (htrack : ∀ t∈S, ∀ N : ℕ, |
| (∀ n : ℕ,n≤N → ‖r.extend.orbit (p+t•V) n‖≤1/3) → |
| (∀ 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‖≤B*ρ^N) : |
| ∀ᶠ t in 𝓝[S] t₀, |
| |H t-H t₀|≤(2*Real.exp (8*(δ : ℝ)/(1-((ℓ+δ : ℝ≥0) : ℝ))))* |t-t₀| := by |
| classical |
| have hcentralP (n : ℕ) : ‖r.extend.orbit (p+t₀•V) n‖≤1/12 := |
| (hp₀ n).trans (div_le_div_of_nonneg_right (pow_le_one₀ hb hb1) (by norm_num)) |
| have hcentralQ (n : ℕ) : ‖r.extend.orbit (q+H t₀•V) n‖≤1/12 := |
| (hq₀ n).trans (div_le_div_of_nonneg_right (pow_le_one₀ hb hb1) (by norm_num)) |
| have hcentralP' (n : ℕ) : ‖r.extend.orbit (p+t₀•V) n‖≤1/3 := (hcentralP n).trans (by norm_num) |
| have hcentralQ' (n : ℕ) : ‖r.extend.orbit (q+H t₀•V) n‖≤1/3 := (hcentralQ n).trans (by norm_num) |
| have hdecay := eventually_scaled_power_small (C := (1:ℝ)/12) (d := 1/(6*M)) ha hb hab (by positivity) |
| have hpair := eventually_scaled_power_small (C := B) (d := 1/(24*M)) ha hρ haρ (by positivity) |
| obtain ⟨N₀,hN₀⟩ := eventually_atTop.mp (hdecay.and hpair) |
| have hpre (n : ℕ) : ∀ᶠ t in 𝓝[S] t₀, |
| ‖r.extend.orbit (p+t•V) n‖≤1/3 ∧ ‖r.extend.orbit (q+H t•V) n‖≤1/3 := by |
| have hp : ContinuousAt (fun t : ℝ => ‖r.extend.orbit (p+t•V) n‖) t₀ := |
| ((r.extend.orbit_continuous n).comp (continuous_const.add (continuous_id.smul continuous_const))).norm.continuousAt |
| have hq : ContinuousWithinAt (fun t : ℝ => ‖r.extend.orbit (q+H t•V) n‖) S t₀ := |
| (r.extend.orbit_continuous n).continuousAt.continuousWithinAt.comp |
| (continuousWithinAt_const.add (hH.smul continuousWithinAt_const)) (mapsTo_univ _ _) |>.norm |
| exact ((hp.eventually (gt_mem_nhds ((hcentralP n).trans_lt (by norm_num)))).filter_mono |
| inf_le_left).and (hq.eventually (gt_mem_nhds ((hcentralQ n).trans_lt (by norm_num)))) |>.mono |
| (fun t ht => ⟨ht.1.le,ht.2.le⟩) |
| have hprefix : ∀ᶠ t in 𝓝[S] t₀, ∀ n∈Finset.range (N₀+1), |
| ‖r.extend.orbit (p+t•V) n‖≤1/3 ∧ ‖r.extend.orbit (q+H t•V) n‖≤1/3 := by |
| exact (Finset.eventually_all _).mpr (fun n hn => hpre n) |
| filter_upwards [hprefix,self_mem_nhdsWithin] with t ht htS |
| by_cases he : t=t₀ |
| · simp [he] |
| have hcone (x y : ℝ) : |(p+x•V).1-(p+y•V).1|≤|(p+x•V).2-(p+y•V).2| := by |
| change |p.1+x*V.1-(p.1+y*V.1)|≤|p.2+x*V.2-(p.2+y*V.2)| |
| have h₁ : p.1+x*V.1-(p.1+y*V.1)=(x-y)*V.1 := by ring |
| have h₂ : p.2+x*V.2-(p.2+y*V.2)=(x-y)*V.2 := by ring |
| rw [h₁,h₂,abs_mul,abs_mul] |
| exact mul_le_mul_of_nonneg_left hc (abs_nonneg _) |
| have hconeQ (x y : ℝ) : |(q+x•V).1-(q+y•V).1|≤|(q+x•V).2-(q+y•V).2| := by |
| change |q.1+x*V.1-(q.1+y*V.1)|≤|q.2+x*V.2-(q.2+y*V.2)| |
| have h₁ : q.1+x*V.1-(q.1+y*V.1)=(x-y)*V.1 := by ring |
| have h₂ : q.2+x*V.2-(q.2+y*V.2)=(x-y)*V.2 := by ring |
| rw [h₁,h₂,abs_mul,abs_mul] |
| exact mul_le_mul_of_nonneg_left hc (abs_nonneg _) |
| have hne : (p+t•V).2≠(p+t₀•V).2 := by |
| intro hh |
| change p.2+t*V.2=p.2+t₀*V.2 at hh |
| exact he (mul_right_cancel₀ hV (add_left_cancel hh)) |
| obtain ⟨j,hj⟩ := r.extend.exists_orbit_escape h (hcone t t₀) hne hcentralP' |
| have hbad : ∃ n : ℕ, ¬(‖r.extend.orbit (p+t•V) n‖≤1/3 ∧ |
| ‖r.extend.orbit (q+H t•V) n‖≤1/3) := ⟨j,fun hh => hj.not_ge hh.1⟩ |
| obtain ⟨N,hN,htr,hout⟩ := last_trapped_time hbad (N₀ := N₀) |
| (fun n hn => ht n (Finset.mem_range.mpr (by omega))) |
| have hcentral (n : ℕ) (hn : N₀≤n) : b^n/12≤1/(6*M*a^n) := by |
| have hh := (hN₀ n hn).1 |
| convert hh using 1 <;> ring |
| have hpairbound : B*ρ^N≤(1/(6*M*a^N))/4 := by |
| have hh := (hN₀ N hN).2 |
| convert hh using 1 |
| ring |
| have hesc : 1/(6*M*a^N)< |(r.extend.orbit (p+t•V) N).2-(r.extend.orbit (p+t₀•V) N).2| ∨ |
| 1/(6*M*a^N)< |(r.extend.orbit (q+H t•V) N).2-(r.extend.orbit (q+H t₀•V) N).2| := by |
| push Not at hout |
| by_cases hp : 1/3<‖r.extend.orbit (p+t•V) (N+1)‖ |
| · exact Or.inl (r.extend.escape_vertical_lower h (hcone t t₀) hM ha hstep hp |
| ((hp₀ N).trans (hcentral N hN))) |
| · exact Or.inr (r.extend.escape_vertical_lower h (hconeQ (H t) (H t₀)) hM ha hstep |
| (hout (le_of_not_gt hp)) ((hq₀ N).trans (hcentral N hN))) |
| have hpairs₀ : |(r.extend.orbit (p+t₀•V) N).2-(r.extend.orbit (q+H t₀•V) N).2|≤(1/(6*M*a^N))/4 := by |
| exact (norm_snd_le _).trans ((htrack t₀ ht₀ N (fun n _ => hcentralP' n) |
| (fun n _ => hcentralQ' n)).trans hpairbound) |
| have hpairs₁ : |(r.extend.orbit (p+t•V) N).2-(r.extend.orbit (q+H t•V) N).2|≤(1/(6*M*a^N))/4 := by |
| exact (norm_snd_le _).trans ((htrack t htS N (fun n hn => (htr n hn).1) |
| (fun n hn => (htr n hn).2)).trans hpairbound) |
| have hterminal := (terminal_size_comparison (by positivity : 0<(1/(6*M*a^N))) |
| hesc hpairs₀ hpairs₁).1 |
| exact r.finite_holonomy_ratio h hδ D hd hsmall p q V hc hV |
| (fun n _ => hcentralP' n) (fun n hn => (htr n hn).1) |
| (fun n _ => hcentralQ' n) (fun n hn => (htr n hn).2) hterminal |
|
|
| end StandardMapEntropy.NonlinearStable |
|
|
| end |
| section |
| namespace StandardMapEntropy |
| open MeasureTheory Set Filter |
| open scoped Topology NNReal ENNReal |
| open NonlinearStable |
|
|
| lemma fine_trueOrbit (k χ ε δ : ℝ) (hδ : 0<δ) (w : ℂ) |
| (hw : ∀ n : ℕ, FineRegular k χ ε (complexProjection ((standardLift k)^[n] w))) |
| (v : RealPlane) (n : ℕ) : |
| (fineLocalRecurrence k χ ε δ hδ w hw).trueOrbit v n= |
| fineCoordinate k χ ε δ w (w+fineFrame k χ ε δ (complexProjection w) v) n := by |
| induction n with |
| | zero => exact (fineCoordinate_initial k χ ε hδ w (hw 0) v).symm |
| | succ n ih => rw [LocalRecurrence.trueOrbit,ih,fineCoordinate_step] |
|
|
| theorem fineTrapped_hasDerivAt_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} {N : ℕ} (htrap : v∈fineTrapped k χ ε δ w N) : |
| ∃ J : ℝ, HasDerivAt (fun t : ℝ => |
| (fineCoordinate k χ ε δ w (w+fineFrame k χ ε δ (complexProjection w) (v.1,t)) N).2) J v.2 ∧ |
| J≠0 ∧ abs (Real.log |J|-(∑ j ∈ Finset.range N,Real.log |fineUnstableMultiplier k χ ε δ |
| (complexProjection ((standardLift k)^[j] w))|))≤4*δ/(1-(Real.exp (-χ+ε)+δ)) := by |
| classical |
| let r := fineLocalRecurrence k χ ε δ hδ w hw |
| let D (j : ℕ) := fineRemainderDerivative k χ ε δ ((standardLift k)^[j] w) |
| let L (j : ℕ) : Plane →L[ℝ] Plane := if j<N then D j (r.trueOrbit v j) else 0 |
| have hv : v∈r.extend.trappedBox N := fineTrapped_subset k χ ε δ hδ w hw N htrap |
| have hLsmall (j : ℕ) (hj : j<N) : ‖L j‖≤δ*‖r.extend.orbit v j‖ := by |
| dsimp only [L] |
| rw [ite_eq_left hj,r.trueOrbit_eq_extend hv hj.le] |
| have hh := fineRemainderDerivative_lipschitz k χ ε hδ ((standardLift k)^[j] w) |
| (hw j) (fine_regular_lift_next k χ ε w hw j) (r.extend.orbit v j) 0 |
| simpa only [fineRemainderDerivative_zero,sub_zero] using hh |
| have hL (j : ℕ) : ‖L j‖≤fineErrorRate δ hδ := by |
| by_cases hj : j<N |
| · exact (hLsmall j hj).trans ((mul_le_mul_of_nonneg_left (hv j hj.le) hδ.le).trans_eq (mul_one _)) |
| · simp only [L,ite_eq_right hj,norm_zero]; exact hδ.le |
| let R := r.linearized L hL |
| refine ⟨(R.orbit (0,1) N).2,?_,?_,?_⟩ |
| · have hd := r.trueOrbit_hasDerivAt D (fun j u => hasFDerivAt_fineRemainder k χ ε δ _ u) |
| L hL (v := v) (N := N) (fun j hj => ite_eq_left hj) (le_refl N) |
| have hs : HasDerivAt (fun t : ℝ => (r.trueOrbit (v.1,t) N).2) (R.orbit (0,1) N).2 v.2 := |
| by exact hd.snd |
| simpa only [r,fine_trueOrbit] using hs |
| · exact R.orbit_snd_ne_zero hq (v := (0,1)) (by norm_num) one_ne_zero N |
| · exact r.linearized_log_distortion hq hδ' L hL hv hLsmall |
|
|
| end StandardMapEntropy |
|
|
| end |
| section |
| namespace StandardMapEntropy |
| open MeasureTheory Set Filter |
| open scoped Topology NNReal ENNReal |
| open NonlinearStable |
|
|
| lemma fineInverse_forward_bound (k χ ε δ : ℝ) (hδ : 0<δ) (w : ℂ) |
| (hw : ∀ n : ℕ, FineRegular k χ ε (complexProjection ((standardLift k)^[n] w))) (n : ℕ) : |
| ‖fineInverse k χ ε δ (complexProjection ((standardLift k)^[n] w))‖≤ |
| (Real.exp (2*ε))^n/(100*codingRadius k χ ε δ (complexProjection w)) := by |
| have hr := codingRadius_pos k χ ε hδ (complexProjection w) |
| have hh := fineInverse_forward_radius_bound k χ ε δ hδ w hw n |
| apply (le_div_iff₀ (by positivity)).mpr |
| nlinarith |
|
|
| lemma fineCoordinate_tracking_rate (k χ ε δ : ℝ) (hδ : 0<δ) (w p p' : ℂ) |
| (hw : ∀ n : ℕ, FineRegular k χ ε (complexProjection ((standardLift k)^[n] w))) |
| {D b : ℝ} (hD : 0≤D) (hb : 0≤b) |
| (htrack : ∀ n : ℕ, ‖(standardLift k)^[n] p-(standardLift k)^[n] p'‖≤D*b^n) (n : ℕ) : |
| ‖fineCoordinate k χ ε δ w p n-fineCoordinate k χ ε δ w p' n‖≤ |
| (D/(100*codingRadius k χ ε δ (complexProjection w)))*(Real.exp (2*ε)*b)^n := by |
| have he : fineCoordinate k χ ε δ w p n-fineCoordinate k χ ε δ w p' n= |
| fineInverse k χ ε δ (complexProjection ((standardLift k)^[n] w)) |
| ((standardLift k)^[n] p-(standardLift k)^[n] p') := by |
| dsimp only [fineCoordinate] |
| rw [←map_sub,sub_sub_sub_cancel_right] |
| rw [he] |
| calc |
| _ ≤ ‖fineInverse k χ ε δ (complexProjection ((standardLift k)^[n] w))‖* |
| ‖(standardLift k)^[n] p-(standardLift k)^[n] p'‖ := ContinuousLinearMap.le_opNorm _ _ |
| _ ≤ ((Real.exp (2*ε))^n/(100*codingRadius k χ ε δ (complexProjection w)))*(D*b^n) := |
| (mul_le_mul_of_nonneg_left (htrack n) (norm_nonneg _)).trans |
| (mul_le_mul_of_nonneg_right (fineInverse_forward_bound k χ ε δ hδ w hw n) |
| (mul_nonneg hD (pow_nonneg hb n))) |
| _ = _ := by rw [mul_pow]; ring |
|
|
| theorem fine_extended_step_bound (k : ℝ) (hk : 0≤k) (χ ε δ : ℝ) (hε : 0≤ε) (hδ : 0<δ) |
| (w : ℂ) (hw : ∀ n : ℕ, FineRegular k χ ε (complexProjection ((standardLift k)^[n] w))) : |
| ∃ M : ℝ, 0<M ∧ ∀ n u, ‖(fineLocalRecurrence k χ ε δ hδ w hw).extend.step n u‖≤ |
| M*(Real.exp (2*ε))^n*‖u‖ := by |
| let a := Real.exp (2*ε) |
| let R := codingRadius k χ ε δ (complexProjection w) |
| let C := (a*(9*growthBase k)*(2*fineScale k ε δ))/(100*R) |
| let r := fineLocalRecurrence k χ ε δ hδ w hw |
| have ha : 1≤a := Real.one_le_exp_iff.mpr (by positivity) |
| have hR : 0<R := codingRadius_pos k χ ε hδ _ |
| have hG : 0<growthBase k := by unfold growthBase; positivity |
| have hC : 0<C := by dsimp [C,a]; positivity [fineScale_pos k ε hδ] |
| refine ⟨C+δ,by positivity,fun n u => ?_⟩ |
| let z := complexProjection ((standardLift k)^[n] w) |
| have hI : ‖fineInverse k χ ε δ (standardMap k z)‖≤a^(n+1)/(100*R) := by |
| simpa only [z,Function.iterate_succ_apply',complexProjection_standardLift] using |
| fineInverse_forward_bound k χ ε δ hδ w hw (n+1) |
| have hF : ‖fineFrame k χ ε δ z‖≤2*fineScale k ε δ := |
| (fineFrame_norm_le k χ ε hδ z).trans (mul_le_mul_of_nonneg_left (fineSize_le_scale k χ ε hδ z) (by norm_num)) |
| have hdiag : ‖((r.a n*u.1,r.b n*u.2) : Plane)‖≤C*a^n*‖u‖ := by |
| rw [show ((r.a n*u.1,r.b n*u.2) : Plane)= |
| fineInverse k χ ε δ (standardMap k z) (standardDerivative k z (fineFrame k χ ε δ z u)) from |
| (fine_derivative_diagonal k χ ε δ z (hw n) u).symm] |
| calc |
| _ ≤ ‖fineInverse k χ ε δ (standardMap k z)‖* |
| (‖standardDerivative k z‖*(‖fineFrame k χ ε δ z‖*‖u‖)) := |
| (ContinuousLinearMap.le_opNorm _ _).trans (mul_le_mul_of_nonneg_left |
| ((ContinuousLinearMap.le_opNorm _ _).trans (mul_le_mul_of_nonneg_left |
| (ContinuousLinearMap.le_opNorm _ _) (norm_nonneg _))) (norm_nonneg _)) |
| _ ≤ (a^(n+1)/(100*R))*((9*growthBase k)*((2*fineScale k ε δ)*‖u‖)) := |
| mul_le_mul hI (mul_le_mul (standardDerivative_norm_bound k hk z) |
| (mul_le_mul_of_nonneg_right hF (norm_nonneg _)) (by positivity) (by positivity)) |
| (by positivity) (by positivity) |
| _ = _ := by dsimp only [C]; rw [pow_succ]; ring |
| have hrem : ‖r.extend.remainder n u‖≤δ*‖u‖ := by |
| have ht := (r.extend.lipschitz n).dist_le_mul u 0 |
| change ‖r.extend.remainder n u-r.extend.remainder n 0‖≤δ*‖u-0‖ at ht |
| rw [r.extend.zero,sub_zero,sub_zero] at ht |
| exact ht |
| have he : r.extend.step n u=((r.a n*u.1,r.b n*u.2) : Plane)+r.extend.remainder n u := rfl |
| change ‖r.extend.step n u‖≤_ |
| rw [he] |
| have hap : 1≤a^n := one_le_pow₀ ha |
| have hh := (norm_add_le _ _).trans (add_le_add hdiag hrem) |
| have hrem' : δ*‖u‖≤δ*a^n*‖u‖ := by |
| have hh := mul_le_mul_of_nonneg_left hap hδ.le |
| simpa only [mul_one] using mul_le_mul_of_nonneg_right hh (norm_nonneg u) |
| exact hh.trans (by nlinarith) |
|
|
| end StandardMapEntropy |
|
|
| end |
| end |
|
|
| end OAI |
|
|
