| import OAI.Dynamics.SmoothObstruction.Splitting |
|
|
| namespace OAI |
|
|
| section |
|
|
| open MeasureTheory Filter Topology TopologicalSpace Set Finset |
| open scoped ENNReal NNReal BoundedContinuousFunction |
| open Finset MeasureTheory Filter Topology |
| open scoped BigOperators |
| open Finset Matrix |
| open scoped Matrix.Norms.L2Operator |
| open Matrix |
|
|
| namespace SmoothObstruction.GoodStationary |
| open Filter Topology |
|
|
| lemma floor_burden_eq (k : ℕ) : |
| floorAt k*(2:ℝ)^k=Real.exp ((9*Real.log 2/10)*(k:ℝ)) := by |
| have hp : (2:ℝ)^k=Real.exp (Real.log 2*(k:ℝ)) := by |
| rw [mul_comm,Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)] |
| rw [floorAt,Real.rpow_def_of_pos (by norm_num : (0:ℝ)<2),hp,←Real.exp_add] |
| congr 1 |
| ring |
|
|
| lemma burden_ge_one (b : ℕ → ℝ) (hb : ∀ k,floorAt k≤b k) (k : ℕ) : |
| 1≤b k*(2:ℝ)^k := by |
| have hc : 0≤9*Real.log 2/10 := by positivity |
| have he := Real.one_le_exp_iff.mpr (mul_nonneg hc (Nat.cast_nonneg k)) |
| rw [←floor_burden_eq] at he |
| exact he.trans (mul_le_mul_of_nonneg_right (hb k) (pow_nonneg (by norm_num) _)) |
|
|
| lemma eventual_log_le_burden (b : ℕ → ℝ) (hb : ∀ k,floorAt k≤b k) : |
| ∀ᶠ k in atTop,Real.log ((2^(k+1):ℕ)+1)≤b k*(2:ℝ)^k := by |
| have hc : 0<9*Real.log 2/10 := by positivity |
| have ho := (isLittleO_pow_exp_pos_mul_atTop 1 hc).comp_tendsto |
| (tendsto_natCast_atTop_atTop : Tendsto (fun k : ℕ => (k:ℝ)) atTop atTop) |
| filter_upwards [ho.bound (show (0:ℝ)<1/3 by norm_num),Filter.eventually_ge_atTop 1] with k hk hk1 |
| simp only [Function.comp_apply,pow_one,Real.norm_eq_abs,abs_of_nonneg (Nat.cast_nonneg k : (0:ℝ)≤k), |
| abs_of_pos (Real.exp_pos _)] at hk |
| have hpow : ((2^(k+1):ℕ):ℝ)+1≤(2:ℝ)^(k+2) := by |
| have hh : (1:ℝ)≤2^(k+1) := one_le_pow₀ (by norm_num) |
| calc |
| _ = (2:ℝ)^(k+1)+1 := by norm_cast |
| _ ≤ (2:ℝ)^(k+1)*2 := by linarith |
| _ = (2:ℝ)^(k+2) := by simp only [show k+2=(k+1)+1 by omega,pow_succ] |
| have hlog := Real.log_le_log (by positivity : (0:ℝ)<((2^(k+1):ℕ):ℝ)+1) hpow |
| rw [Real.log_pow] at hlog |
| have h2 : Real.log 2≤1 := by |
| have hh := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2) |
| linarith |
| have hkk : (k:ℝ)+2≤3*k := by exact_mod_cast (show k+2≤3*k by omega) |
| have hlog' : Real.log (((2^(k+1):ℕ):ℝ)+1)≤3*k := by |
| exact hlog.trans ((mul_le_mul_of_nonneg_left h2 (by positivity)).trans (by simpa using hkk)) |
| have he : 3*(k:ℝ)≤floorAt k*2^k := by rw [floor_burden_eq]; linarith |
| exact hlog'.trans (he.trans (mul_le_mul_of_nonneg_right (hb k) (by positivity))) |
| end SmoothObstruction.GoodStationary |
|
|
| namespace SmoothObstruction.OrbitCocycle |
| open MeasureTheory Filter Topology Set |
| open scoped Matrix.Norms.L2Operator |
| variable {X : Type*} [MetricSpace X] [CompactSpace X] [MeasurableSpace X] [BorelSpace X] |
| {d : ℕ} {K : ℝ} |
|
|
| omit [CompactSpace X] [MeasurableSpace X] [BorelSpace X] in |
| lemma code_iterate (S : X ≃ₜ X) (A : X → CompactMatrix.Alphabet d K) (x : X) (t : ℕ) : |
| shift^[t] (code S A x)=code S A (S^[t] x) := by |
| induction t with |
| | zero => rfl |
| | succ t ih => |
| rw [Function.iterate_succ_apply',ih,←code_shift,Function.iterate_succ_apply'] |
|
|
| omit [CompactSpace X] [MeasurableSpace X] [BorelSpace X] in |
| lemma matrix_at_actual (S : X ≃ₜ X) (A : X → CompactMatrix.Alphabet d K) |
| (p : Path (X:=X) (d:=d) (K:=K)) (hp : p∈Actual S A) (t : ℕ) : |
| (matrix (shift^[t] p)).toContinuousLinearMap=(A (S^[t] (p 0).1)).val := by |
| have he : shift^[t] p=code S A (S^[t] (p 0).1) := by |
| conv_lhs => rw [show p=code S A (p 0).1 from hp] |
| exact code_iterate S A (p 0).1 t |
| rw [he] |
| simp only [matrix,code,zpow_zero,Homeomorph.one_apply] |
| rfl |
|
|
| / |
| The compact stationary QR lift, square-summable burdens, compact good |
| sets, and their original base masses are all constructed. On every good |
| witness the operator identity uses the ACTUAL original cocycle, not a |
| free triangular surrogate or a separately assumed frame process. -/ |
| theorem actual_good_gauges (S : X ≃ₜ X) (A : X → CompactMatrix.Alphabet d K) |
| (hA : Measurable A) (μ : ProbabilityMeasure X) |
| (hS : MeasurePreserving S (μ : Measure X) (μ : Measure X)) |
| {η : ℝ} (hη : 0<η) : |
| ∃ (ν : ProbabilityMeasure (Path (X:=X) (d:=d) (K:=K) × QR.Frame d)) |
| (b : ℕ → ℝ) (G : ℕ → Set (Path (X:=X) (d:=d) (K:=K) × QR.Frame d)) |
| (R C : ℝ), |
| 0≤R ∧ 0≤C ∧ (∀ k,0<b k) ∧ (∀ k,GoodStationary.floorAt k≤b k) ∧ |
| Summable (fun k => (b k)^2) ∧ |
| (∀ k,IsCompact (G k) ∧ |
| (∀ z∈G k,z.1∈Actual S A) ∧ |
| 1-η/100≤(ν : Measure _).real (G k) ∧ |
| IsCompact ((fun z => (z.1 ((2^k : ℕ) : ℤ)).1) '' G k) ∧ |
| 1-η/100≤(μ : Measure X).real ((fun z => (z.1 ((2^k : ℕ) : ℤ)).1) '' G k)) ∧ |
| ∀ᶠ k in atTop,∀ z∈G k, |
| ∃ (lam : Fin d → ℝ) (P Q : ℕ → Matrix (Fin d) (Fin d) ℝ), |
| (∀ j,|lam j|≤R) ∧ (∀ t,P t*Q t=1 ∧ Q t*P t=1) ∧ |
| (∀ t≤2^(k+1),‖P t‖≤Real.exp (C*(b k*2^k))) ∧ |
| (∀ t≤2^(k+1),‖Q t‖≤Real.exp (C*(b k*2^k))) ∧ |
| (∀ t<2^(k+1), |
| (Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm |
| (A (S^[t] (z.1 0).1)).val*P t= |
| P (t+1)*EuclideanSplitting.rate lam 1) := by |
| let := TopologicalSpace.metrizableSpaceMetric (Path (X:=X) (d:=d) (K:=K)) |
| obtain ⟨ν,b,G,hbp,hbf,hbs,hG⟩ := compact_actual_good S A hA μ hS hη |
| obtain ⟨R,C,hR,hC,hg⟩ := QR.compact_good_gauge |
| (shift (X:=X) (d:=d) (K:=K)) matrix continuous_matrix |
| refine ⟨ν,b,G,R,C,hR,hC,hbp,hbf,hbs,fun k => ?_,?_⟩ |
| · exact ⟨(hG k).1,fun z hz => ((hG k).2.1 z hz).1,(hG k).2.2⟩ |
| · filter_upwards [GoodStationary.eventual_log_le_burden b hbf] with k hk |
| intro z hz |
| obtain ⟨lam,P,Q,hlam,hPQ,hP,hQ,hEq⟩ := hg k (b k) z ((hG k).2.1 z hz).2 |
| (GoodStationary.burden_ge_one b hbf k) hk |
| refine ⟨lam,P,Q,hlam,hPQ,hP,hQ,fun t ht => ?_⟩ |
| have ha := ((hG k).2.1 z hz).1 |
| simpa only [matrix_at_actual S A z.1 ha] using hEq t ht |
| end SmoothObstruction.OrbitCocycle |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Bundle Set Filter Topology MeasureTheory |
| open scoped Manifold ContDiff Matrix.Norms.L2Operator |
| universe u |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type u} [MetricSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
| [CompactSpace M] [MeasurableSpace M] [BorelSpace M] |
|
|
| / |
| smoothness, compactness and the invariant original probability. The finite |
| chart selector, stationary frame lift, summable error sequence, large |
| compact middle-point sets and two-sided invertible gauges are all produced. |
| This is the linear input to the nonlinear switching proof, not that proof. -/ |
| theorem manifold_good_gauges_discrete (S : Diffeomorph I I M M ∞) |
| (μ : ProbabilityMeasure M) (hS : MeasurePreserving S (μ : Measure M) (μ : Measure M)) |
| {η : ℝ} (hη : 0<η) : |
| ∃ (ι : Type u) (_ : Fintype ι) (_ : MeasurableSpace ι) (_ : MeasurableSingletonClass ι) |
| (a : ι → M) (K : ι → Set M) (c : M → ι) (D : ℝ) |
| (A : M → CompactMatrix.Alphabet d D), |
| (∀ i,IsCompact (K i)) ∧ (∀ i,K i⊆(chartAt H (a i)).source) ∧ |
| Measurable c ∧ (∀ x,x∈K (c x)) ∧ 0<D ∧ Measurable A ∧ |
| (∀ x,(A x).val=coord I S (a (c x)) (a (c (S x))) x) ∧ |
| ∃ (ν : ProbabilityMeasure (OrbitCocycle.Path (X:=M) (d:=d) (K:=D) × QR.Frame d)) |
| (b : ℕ → ℝ) (G : ℕ → Set (OrbitCocycle.Path (X:=M) (d:=d) (K:=D) × QR.Frame d)) |
| (R C : ℝ), |
| 0≤R ∧ 0≤C ∧ (∀ k,0<b k) ∧ (∀ k,GoodStationary.floorAt k≤b k) ∧ |
| Summable (fun k => (b k)^2) ∧ |
| (∀ k,IsCompact (G k) ∧ |
| (∀ z∈G k,z.1∈OrbitCocycle.Actual S.toHomeomorph A) ∧ |
| 1-η/100≤(ν : Measure _).real (G k) ∧ |
| IsCompact ((fun z => (z.1 ((2^k : ℕ) : ℤ)).1) '' G k) ∧ |
| 1-η/100≤(μ : Measure M).real ((fun z => (z.1 ((2^k : ℕ) : ℤ)).1) '' G k)) ∧ |
| ∀ᶠ k in atTop,∀ z∈G k, |
| ∃ (lam : Fin d → ℝ) (P Q : ℕ → Matrix (Fin d) (Fin d) ℝ), |
| (∀ j,|lam j|≤R) ∧ (∀ t,P t*Q t=1 ∧ Q t*P t=1) ∧ |
| (∀ t≤2^(k+1),‖P t‖≤Real.exp (C*(b k*2^k))) ∧ |
| (∀ t≤2^(k+1),‖Q t‖≤Real.exp (C*(b k*2^k))) ∧ |
| (∀ t<2^(k+1),let x := S^[t] (z.1 0).1 |
| (Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm |
| (coord I S (a (c x)) (a (c (S x))) x)*P t= |
| P (t+1)*EuclideanSplitting.rate lam 1) := by |
| obtain ⟨ι,hfin,hmeas,hsc,a,K,c,D,A,hK,hKU,hc,hcK,hD,hA,heq⟩ := actual_coordinate_cocycle_discrete I S |
| let : Fintype ι := hfin |
| let : MeasurableSpace ι := hmeas |
| obtain ⟨ν,b,G,R,C,hR,hC,hbp,hbf,hbs,hG,hg⟩ := |
| OrbitCocycle.actual_good_gauges S.toHomeomorph A hA μ hS hη |
| refine ⟨ι,hfin,hmeas,hsc,a,K,c,D,A,hK,hKU,hc,hcK,hD,hA,heq, |
| ν,b,G,R,C,hR,hC,hbp,hbf,hbs,hG,?_⟩ |
| filter_upwards [hg] with k hk |
| intro z hz |
| obtain ⟨lam,P,Q,hlam,hPQ,hP,hQ,hEq⟩ := hk z hz |
| refine ⟨lam,P,Q,hlam,hPQ,hP,hQ,fun t ht => ?_⟩ |
| simpa only [heq,Diffeomorph.coe_toHomeomorph] using hEq t ht |
| end SmoothObstruction.ManifoldCoordinates |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Bundle Set Filter Topology |
| open scoped Manifold ContDiff |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
|
|
| / |
| lemma coord_inverse_step (S : Diffeomorph I I M M ∞) (a b x : M) |
| (hx : x∈(chartAt H a).source) (hy : S x∈(chartAt H b).source) : |
| coord I S.symm b a (S x)*coord I S a b x=1 := by |
| have hi : (mfderiv I I S.symm (S x)).comp (mfderiv I I S x)= |
| ContinuousLinearMap.id ℝ (TangentSpace I x) := by |
| rw [←mfderiv_comp x (S.symm.mdifferentiable (by simp) _) (S.mdifferentiable (by simp) _)] |
| have he : ((S.symm : M → M) ∘ (S : M → M))=id := by |
| funext y |
| exact S.symm_apply_apply y |
| rw [he,mfderiv_id] |
| rw [coord,coord] |
| rw [ContinuousLinearMap.inCoordinates_eq hy (by simpa using hx), |
| ContinuousLinearMap.inCoordinates_eq hx hy] |
| ext v : 1 |
| simp only [mul_apply_eq_comp,ContinuousLinearMap.comp_apply, |
| ContinuousLinearEquiv.coe_coe,ContinuousLinearEquiv.symm_apply_apply] |
| have hv := congrArg (fun f => f (((trivializationAt (E (d:=d)) (TangentSpace I) a).continuousLinearEquivAt ℝ x hx).symm v)) hi |
| change (mfderiv I I S.symm (S x)) ((mfderiv I I S x) |
| (((trivializationAt (E (d:=d)) (TangentSpace I) a).continuousLinearEquivAt ℝ x hx).symm v))= |
| ((trivializationAt (E (d:=d)) (TangentSpace I) a).continuousLinearEquivAt ℝ x hx).symm v at hv |
| have hh := congrArg (fun w => ((trivializationAt (E (d:=d)) (TangentSpace I) a).continuousLinearEquivAt ℝ x hx) w) hv |
| rw [ContinuousLinearEquiv.apply_symm_apply] at hh |
| convert hh using 1 |
| · change ((trivializationAt (E (d:=d)) (TangentSpace I) a) |
| (⟨S.symm (S x),_⟩ : TotalSpace (E (d:=d)) (TangentSpace I))).2= |
| ((trivializationAt (E (d:=d)) (TangentSpace I) a) |
| (⟨x,_⟩ : TotalSpace (E (d:=d)) (TangentSpace I))).2 |
| congr 2 |
| exact Bundle.TotalSpace.ext (S.symm_apply_apply x) HEq.rfl |
| · rfl |
| end SmoothObstruction.ManifoldCoordinates |
|
|
| namespace SmoothObstruction.OrbitCocycle |
| open Set |
| variable {X : Type*} [MetricSpace X] {d : ℕ} {K : ℝ} |
|
|
| lemma homeomorph_pow_iterate (S : X ≃ₜ X) (x : X) (n : ℕ) : |
| (S^n) x=S^[n] x := by |
| induction n with |
| | zero => rfl |
| | succ n ih => |
| rw [pow_succ',Homeomorph.mul_apply,Function.iterate_succ_apply',ih] |
|
|
| lemma actual_middle (S : X ≃ₜ X) (A : X → CompactMatrix.Alphabet d K) |
| (p : Path (X:=X) (d:=d) (K:=K)) (hp : p∈Actual S A) (n : ℕ) : |
| (p (n:ℤ)).1=S^[n] (p 0).1 := by |
| conv_lhs => rw [show p=code S A (p 0).1 from hp] |
| simp only [code,zpow_natCast] |
| exact homeomorph_pow_iterate S _ n |
|
|
| lemma backward_center (S : X ≃ₜ X) (x : X) {n t : ℕ} (ht : t≤n) : |
| S.symm^[t] (S^[n] x)=S^[n-t] x := by |
| have he : n=t+(n-t) := by omega |
| conv_lhs => rw [he,Function.iterate_add_apply] |
| exact (show Function.LeftInverse S.symm S from S.symm_apply_apply).iterate t _ |
| end SmoothObstruction.OrbitCocycle |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Bundle Set Filter Topology |
| open scoped Manifold ContDiff Matrix.Norms.L2Operator |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
|
|
| / |
| rates are the negatives of the forward rates, and the same center basis |
| is used. No second derivative of a long iterate is introduced. -/ |
| lemma reverse_gauge_step (S : Diffeomorph I I M M ∞) |
| (c : ℕ → M) (x : ℕ → M) (P : ℕ → Matrix (Fin d) (Fin d) ℝ) |
| (lam : Fin d → ℝ) (j : ℕ) |
| (hx : S (x j)=x (j+1)) |
| (hsrc : x j∈(chartAt H (c j)).source) |
| (htar : x (j+1)∈(chartAt H (c (j+1))).source) |
| (hstep : (Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm |
| (coord I S (c j) (c (j+1)) (x j))*P j= |
| P (j+1)*EuclideanSplitting.rate lam 1) : |
| (Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm |
| (coord I S.symm (c (j+1)) (c j) (x (j+1)))*P (j+1)= |
| P j*EuclideanSplitting.rate (fun i => -lam i) 1 := by |
| let F := Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ) |
| let A := F.symm (coord I S (c j) (c (j+1)) (x j)) |
| let Z := F.symm (coord I S.symm (c (j+1)) (c j) (x (j+1))) |
| have hZA : Z*A=1 := by |
| rw [←map_mul,←hx,coord_inverse_step I S _ _ _ hsrc (by rwa [hx]),map_one] |
| have hrate : EuclideanSplitting.rate lam 1* |
| EuclideanSplitting.rate (fun i => -lam i) 1=1 := by |
| have he : EuclideanSplitting.rate (fun i => -lam i) 1= |
| EuclideanSplitting.rate lam (-1) := by simp [EuclideanSplitting.rate] |
| rw [he,←EuclideanSplitting.rate_add] |
| norm_num |
| change Z*P (j+1)=_ |
| calc |
| Z*P (j+1)=(Z*P (j+1))* |
| (EuclideanSplitting.rate lam 1*EuclideanSplitting.rate (fun i => -lam i) 1) := by |
| rw [hrate,mul_one] |
| _=Z*(P (j+1)*EuclideanSplitting.rate lam 1)* |
| EuclideanSplitting.rate (fun i => -lam i) 1 := by noncomm_ring |
| _=Z*(A*P j)*EuclideanSplitting.rate (fun i => -lam i) 1 := by rw [←hstep] |
| _=P j*EuclideanSplitting.rate (fun i => -lam i) 1 := by |
| rw [←mul_assoc Z A,hZA,one_mul] |
| end SmoothObstruction.ManifoldCoordinates |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Bundle Set Filter Topology MeasureTheory |
| open scoped Manifold ContDiff Matrix.Norms.L2Operator |
| universe u |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type u} [MetricSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
|
|
| / |
| This carries no splitting, threshold, switch, Jacobian or measure conclusion. -/ |
| structure BilateralGauge (S : Diffeomorph I I M M ∞) {ι : Type*} |
| (a : ι → M) (c : M → ι) (n : ℕ) (B R : ℝ) (x : M) where |
| lam : Fin d → ℝ |
| rate_bound : ∀ i,|lam i|≤R |
| UF : ℕ → Matrix (Fin d) (Fin d) ℝ |
| VF : ℕ → Matrix (Fin d) (Fin d) ℝ |
| UB : ℕ → Matrix (Fin d) (Fin d) ℝ |
| VB : ℕ → Matrix (Fin d) (Fin d) ℝ |
| invF : ∀ t,UF t*VF t=1 ∧ VF t*UF t=1 |
| invB : ∀ t,UB t*VB t=1 ∧ VB t*UB t=1 |
| centerU : UB 0=UF 0 |
| centerV : VB 0=VF 0 |
| normF : ∀ t≤n,‖UF t‖≤Real.exp B ∧ ‖VF t‖≤Real.exp B |
| normB : ∀ t≤n,‖UB t‖≤Real.exp B ∧ ‖VB t‖≤Real.exp B |
| stepF : ∀ t<n,(Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm |
| (coord I S (a (c (S^[t] x))) (a (c (S^[t+1] x))) (S^[t] x))*UF t= |
| UF (t+1)*EuclideanSplitting.rate lam 1 |
| stepB : ∀ t<n,(Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm |
| (coord I S.symm (a (c (S.symm^[t] x))) (a (c (S.symm^[t+1] x))) (S.symm^[t] x))*UB t= |
| UB (t+1)*EuclideanSplitting.rate (fun i => -lam i) 1 |
|
|
| lemma center_from_gauge (S : Diffeomorph I I M M ∞) {ι : Type*} |
| (a : ι → M) (c : M → ι) (hc : ∀ x,x∈(chartAt H (a (c x))).source) |
| (x₀ : M) (n : ℕ) (B R : ℝ) |
| (lam : Fin d → ℝ) (P Q : ℕ → Matrix (Fin d) (Fin d) ℝ) |
| (hlam : ∀ i,|lam i|≤R) (hPQ : ∀ t,P t*Q t=1 ∧ Q t*P t=1) |
| (hb : ∀ t≤n+n,‖P t‖≤Real.exp B ∧ ‖Q t‖≤Real.exp B) |
| (hstep : ∀ t<n+n,(Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm |
| (coord I S (a (c (S^[t] x₀))) (a (c (S^[t+1] x₀))) (S^[t] x₀))*P t= |
| P (t+1)*EuclideanSplitting.rate lam 1) : |
| Nonempty (BilateralGauge I S a c n B R (S^[n] x₀)) := by |
| have hf (t : ℕ) : S^[t] (S^[n] x₀)=S^[n+t] x₀ := by |
| rw [Nat.add_comm n t,Function.iterate_add_apply] |
| have hr (t : ℕ) (ht : t≤n) : S.symm^[t] (S^[n] x₀)=S^[n-t] x₀ := |
| OrbitCocycle.backward_center S.toHomeomorph x₀ ht |
| refine ⟨{ |
| lam := lam |
| rate_bound := hlam |
| UF := fun t => P (n+t) |
| VF := fun t => Q (n+t) |
| UB := fun t => P (n-t) |
| VB := fun t => Q (n-t) |
| invF := fun t => hPQ (n+t) |
| invB := fun t => hPQ (n-t) |
| centerU := by simp |
| centerV := by simp |
| normF := fun t ht => hb (n+t) (by omega) |
| normB := fun t ht => hb (n-t) (by omega) |
| stepF := ?_ |
| stepB := ?_ |
| }⟩ |
| · intro t ht |
| simp only [hf,←Nat.add_assoc] |
| exact hstep (n+t) (by omega) |
| · intro t ht |
| rw [hr t ht.le,hr (t+1) (by omega)] |
| have he : n-(t+1)+1=n-t := by omega |
| have hh := reverse_gauge_step I S (fun j => a (c (S^[j] x₀))) |
| (fun j => S^[j] x₀) P lam (n-(t+1)) |
| (by simp only [Function.iterate_succ_apply']) (hc _) (hc _) |
| (hstep _ (by omega)) |
| simpa only [he] using hh |
| |
| /-- The compact, large original-measure good base sets now carry actual |
| forward AND backward gauges about their own time-zero point. The QR path |
| witness is eliminated existentially; no measurable frame selection is used. -/ |
| theorem good_bilateral_gauges [CompactSpace M] [MeasurableSpace M] [BorelSpace M] |
| (S : Diffeomorph I I M M ∞) (μ : ProbabilityMeasure M) |
| (hS : MeasurePreserving S (μ : Measure M) (μ : Measure M)) {η : ℝ} (hη : 0<η) : |
| ∃ (ι : Type u) (_ : Fintype ι) (_ : MeasurableSpace ι) (_ : MeasurableSingletonClass ι) |
| (a : ι → M) (K : ι → Set M) (c : M → ι), |
| (∀ i,IsCompact (K i)) ∧ (∀ i,K i⊆(chartAt H (a i)).source) ∧ |
| Measurable c ∧ (∀ x,x∈K (c x)) ∧ |
| ∃ (b : ℕ → ℝ) (R C : ℝ) (G : ℕ → Set M), |
| 0≤R ∧ 0≤C ∧ (∀ k,0<b k) ∧ (∀ k,GoodStationary.floorAt k≤b k) ∧ |
| Summable (fun k => (b k)^2) ∧ |
| (∀ k,IsCompact (G k) ∧ 1-η/100≤(μ : Measure M).real (G k)) ∧ |
| ∀ᶠ k in atTop,∀ x∈G k, |
| Nonempty (BilateralGauge I S a c (2^k) (C*(b k*2^k)) R x) := by |
| obtain ⟨ι,hfin,hmeas,hsc,a,K,c,D,A,hK,hKU,hc,hcK,hD,hA,heq, |
| ν,b,F,R,C,hR,hC,hbp,hbf,hbs,hF,hg⟩ := manifold_good_gauges_discrete I S μ hS hη |
| let : Fintype ι := hfin |
| let : MeasurableSpace ι := hmeas |
| let G (k : ℕ) := (fun z : OrbitCocycle.Path (X:=M) (d:=d) (K:=D) × QR.Frame d => |
| (z.1 ((2^k:ℕ):ℤ)).1) '' F k |
| refine ⟨ι,hfin,hmeas,hsc,a,K,c,hK,hKU,hc,hcK,b,R,C,G,hR,hC,hbp,hbf,hbs, |
| fun k => ⟨(hF k).2.2.2.1,(hF k).2.2.2.2⟩,?_⟩ |
| filter_upwards [hg] with k hk |
| rintro x ⟨z,hz,rfl⟩ |
| obtain ⟨lam,P,Q,hlam,hPQ,hP,hQ,hstep⟩ := hk z hz |
| have he := OrbitCocycle.actual_middle S.toHomeomorph A z.1 ((hF k).2.1 z hz) (2^k) |
| change (z.1 ((2^k:ℕ):ℤ)).1=S^[2^k] (z.1 0).1 at he |
| dsimp only |
| rw [he] |
| have hn : (2^k:ℕ)+2^k=2^(k+1) := by rw [pow_succ]; omega |
| apply center_from_gauge I S a c (fun x => hKU _ (hcK x)) (z.1 0).1 (2^k) |
| (C*(b k*2^k)) R lam P Q hlam hPQ |
| · intro t ht |
| exact ⟨hP t (by rwa [←hn]),hQ t (by rwa [←hn])⟩ |
| · intro t ht |
| have hh := hstep t (by rwa [←hn]) |
| simpa only [Function.iterate_succ_apply'] using hh |
| end SmoothObstruction.ManifoldCoordinates |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Bundle Set Filter Topology Metric |
| open scoped Manifold ContDiff |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
|
|
| noncomputable def chartChange (a b x : M) (ha : x∈(chartAt H a).source) |
| (hb : x∈(chartAt H b).source) : E (d:=d) ≃L[ℝ] E (d:=d) := |
| ((trivializationAt (E (d:=d)) (TangentSpace I) a).continuousLinearEquivAt ℝ x ha).symm.trans |
| ((trivializationAt (E (d:=d)) (TangentSpace I) b).continuousLinearEquivAt ℝ x hb) |
|
|
| lemma chartChange_coe (a b x : M) (ha : x∈(chartAt H a).source) |
| (hb : x∈(chartAt H b).source) : |
| (chartChange I a b x ha hb : E (d:=d) →L[ℝ] E (d:=d))= |
| coord I (Diffeomorph.refl I M ∞) a b x := by |
| rw [coord,ContinuousLinearMap.inCoordinates_eq ha hb] |
| simp only [Diffeomorph.coe_refl,mfderiv_id] |
| rfl |
|
|
| / |
| This is a relation between actual manifold derivatives, not a formal |
| similarity hypothesis supplied in place of them. -/ |
| theorem coord_chart_change (S : Diffeomorph I I M M ∞) (a b a' b' x : M) |
| (hx : x∈domain I S a b) (hx' : x∈domain I S a' b') : |
| coord I S a' b' x= |
| (chartChange I b b' (S x) hx.2 hx'.2 : E (d:=d) →L[ℝ] E (d:=d)) * |
| coord I S a b x * |
| (chartChange I a' a x hx'.1 hx.1 : E (d:=d) →L[ℝ] E (d:=d)) := by |
| rw [←coordEquiv_coe I S a b x hx,←coordEquiv_coe I S a' b' x hx'] |
| ext v |
| simp only [mul_apply_eq_comp,ContinuousLinearEquiv.coe_coe, |
| coordEquiv,chartChange,ContinuousLinearEquiv.trans_apply] |
| simp |
|
|
| lemma chartChange_inverse (a b x : M) (ha : x∈(chartAt H a).source) |
| (hb : x∈(chartAt H b).source) : |
| (chartChange I a b x ha hb).symm=chartChange I b a x hb ha := by |
| ext v |
| simp [chartChange] |
|
|
| end SmoothObstruction.ManifoldCoordinates |
|
|
| namespace SmoothObstruction.CarriedCharts |
| open Set Metric |
| variable {M : Type*} [MetricSpace M] {ι : Type*} [Fintype ι] |
|
|
| lemma finite_positive_lower (r : ι → ℝ) (hr : ∀ i,0 < r i) : |
| ∃ a : ℝ,0 < a ∧ ∀ i,a ≤ r i := by |
| classical |
| have hh (s : Finset ι) : ∃ a : ℝ,0 < a ∧ ∀ i∈s,a ≤ r i := by |
| induction s using Finset.induction_on with |
| | empty => exact ⟨1,by norm_num,by simp⟩ |
| | @insert i s hi ih => |
| obtain ⟨a,ha,has⟩ := ih |
| refine ⟨min a (r i),lt_min ha (hr i),?_⟩ |
| intro j hj |
| rcases Finset.mem_insert.mp hj with hj|hj |
| · subst j; exact min_le_right _ _ |
| · exact (min_le_left _ _).trans (has j hj) |
| obtain ⟨a,ha,hh⟩ := hh Finset.univ |
| exact ⟨a,ha,fun i => hh i (Finset.mem_univ i)⟩ |
|
|
| / |
| Inputs need not stay in the selected compact patches: the neighboring |
| orbit only needs to stay inside this fixed radius of the old orbit. -/ |
| theorem uniform_domains (S : M → M) (hS : Continuous S) |
| (K U : ι → Set M) (hK : ∀ i,IsCompact (K i)) (hU : ∀ i,IsOpen (U i)) |
| (hKU : ∀ i,K i⊆U i) : |
| ∃ r : ℝ,0 < r ∧ ∀ i j x,x∈K i → S x∈K j → |
| closedBall x r⊆U i∩S ⁻¹' U j := by |
| have hex (ij : ι × ι) : ∃ r : ℝ,0 < r ∧ |
| cthickening r (K ij.1∩S ⁻¹' K ij.2)⊆U ij.1∩S ⁻¹' U ij.2 := by |
| apply ((hK ij.1).inter_right ((hK ij.2).isClosed.preimage hS)).exists_cthickening_subset_open |
| · exact (hU ij.1).inter ((hU ij.2).preimage hS) |
| · intro x hx |
| exact ⟨hKU _ hx.1,hKU _ hx.2⟩ |
| choose r hr hrU using hex |
| obtain ⟨a,ha,har⟩ := finite_positive_lower r hr |
| refine ⟨a,ha,fun i j x hx hSx y hy => ?_⟩ |
| exact hrU (i,j) (closedBall_subset_cthickening (E := K i∩S ⁻¹' K j) ⟨hx,hSx⟩ (r (i,j)) |
| (closedBall_subset_closedBall (har (i,j)) hy)) |
|
|
| / |
| connectedness, or absence of a boundary was used in this construction. -/ |
| theorem uniform_bilateral_domains (S : M ≃ₜ M) |
| (K U : ι → Set M) (hK : ∀ i,IsCompact (K i)) (hU : ∀ i,IsOpen (U i)) |
| (hKU : ∀ i,K i⊆U i) : |
| ∃ r : ℝ,0 < r ∧ |
| (∀ i j x,x∈K i → S x∈K j → closedBall x r⊆U i∩S ⁻¹' U j) ∧ |
| (∀ i j x,x∈K i → S.symm x∈K j → closedBall x r⊆U i∩S.symm ⁻¹' U j) := by |
| obtain ⟨a,ha,haf⟩ := uniform_domains S S.continuous K U hK hU hKU |
| obtain ⟨b,hb,hbf⟩ := uniform_domains S.symm S.symm.continuous K U hK hU hKU |
| refine ⟨min a b,lt_min ha hb,?_,?_⟩ |
| · exact fun i j x hx hSx => (closedBall_subset_closedBall (min_le_left a b)).trans |
| (haf i j x hx hSx) |
| · exact fun i j x hx hSx => (closedBall_subset_closedBall (min_le_right a b)).trans |
| (hbf i j x hx hSx) |
|
|
| section Manifold |
| variable {H : Type*} [TopologicalSpace H] [ChartedSpace H M] |
|
|
| / |
| producer. The selector may be discontinuous: no continuity is asserted or |
| needed for the selected cocycle. Comparisons use its fixed CARRIED chart. -/ |
| theorem actual_chart_margins (S : M ≃ₜ M) (a : ι → M) (K : ι → Set M) |
| (hK : ∀ i,IsCompact (K i)) (hKU : ∀ i,K i⊆(chartAt H (a i)).source) |
| (c : M → ι) (hc : ∀ x,x∈K (c x)) : |
| ∃ r : ℝ,0 < r ∧ ∀ x y,dist y x ≤ r → |
| y∈(chartAt H (a (c x))).source ∧ |
| S y∈(chartAt H (a (c (S x)))).source ∧ |
| S.symm y∈(chartAt H (a (c (S.symm x)))).source := by |
| obtain ⟨r,hr,hf,hb⟩ := uniform_bilateral_domains S K |
| (fun i => (chartAt H (a i)).source) hK (fun i => (chartAt H (a i)).open_source) hKU |
| refine ⟨r,hr,fun x y hxy => ?_⟩ |
| have h₁ := hf (c x) (c (S x)) x (hc x) (hc (S x)) hxy |
| have h₂ := hb (c x) (c (S.symm x)) x (hc x) (hc (S.symm x)) hxy |
| exact ⟨h₁.1,h₁.2,h₂.2⟩ |
| end Manifold |
| end SmoothObstruction.CarriedCharts |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Bundle Set Filter Topology Metric |
| open scoped Manifold ContDiff |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [MetricSpace M] [ProperSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
| {ι : Type*} [Fintype ι] |
|
|
| / |
| compact patches. Thus gauges may be transported from a discontinuously |
| selected atlas to a carried atlas without adding point/scale-dependent |
| constants. These are derivatives of the identity on M, also at its boundary. -/ |
| theorem uniform_chart_changes (a : ι → M) (K : ι → Set M) |
| (hK : ∀ i,IsCompact (K i)) (hKU : ∀ i,K i⊆(chartAt H (a i)).source) : |
| ∃ r D : ℝ,0 < r ∧ 1 ≤ D ∧ ∀ i j x, |
| x∈cthickening r (K i) → x∈cthickening r (K j) → |
| x∈(chartAt H (a i)).source ∧ x∈(chartAt H (a j)).source ∧ |
| ‖coord I (Diffeomorph.refl I M ∞) (a i) (a j) x‖ ≤ D ∧ |
| ‖coord I (Diffeomorph.refl I M ∞) (a j) (a i) x‖ ≤ D := by |
| have hex (i : ι) : ∃ r : ℝ,0 < r ∧ cthickening r (K i)⊆(chartAt H (a i)).source := |
| (hK i).exists_cthickening_subset_open (chartAt H (a i)).open_source (hKU i) |
| choose r hr hrU using hex |
| obtain ⟨r₀,hr₀,hr₀r⟩ := CarriedCharts.finite_positive_lower r hr |
| let N (i : ι) := cthickening (r i) (K i) |
| have hNC (i : ι) : IsCompact (N i) := (hK i).cthickening |
| have hb (ij : ι×ι) : ∃ D : ℝ,∀ x∈N ij.1∩N ij.2, |
| ‖coord I (Diffeomorph.refl I M ∞) (a ij.1) (a ij.2) x‖ ≤ D := |
| ((hNC ij.1).inter (hNC ij.2)).exists_bound_of_continuousOn |
| ((continuousOn_coord I (Diffeomorph.refl I M ∞) (a ij.1) (a ij.2)).mono |
| (fun x hx => ⟨hrU _ hx.1,hrU _ hx.2⟩)) |
| choose C hC using hb |
| let D : ℝ := 1+∑ ij,|C ij| |
| have hD : 1 ≤ D := by |
| dsimp [D] |
| exact le_add_of_nonneg_right (Finset.sum_nonneg (fun _ _ => abs_nonneg _)) |
| have hCD (ij : ι×ι) : C ij ≤ D := by |
| have hh := Finset.single_le_sum (fun z _ => abs_nonneg (C z)) (Finset.mem_univ ij) |
| dsimp [D] |
| linarith [le_abs_self (C ij)] |
| refine ⟨r₀,D,hr₀,hD,fun i j x hxi hxj => ?_⟩ |
| have hi : x∈N i := cthickening_mono (hr₀r i) (K i) hxi |
| have hj : x∈N j := cthickening_mono (hr₀r j) (K j) hxj |
| exact ⟨hrU i hi,hrU j hj,(hC (i,j) x ⟨hi,hj⟩).trans (hCD (i,j)), |
| (hC (j,i) x ⟨hj,hi⟩).trans (hCD (j,i))⟩ |
| end SmoothObstruction.ManifoldCoordinates |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Bundle Set Filter Topology Metric |
| open scoped Manifold ContDiff Matrix.Norms.L2Operator |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [MetricSpace M] [ProperSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
| {ι : Type*} [Fintype ι] |
|
|
| / |
| carried chart sequences lying in the enlarged finite atlas. All norm losses |
| are the same fixed D; the actual manifold-derivative conjugation identity |
| is proved above, and is not an extra cocycle assumption. -/ |
| theorem uniform_carried_gauge (a : ι → M) (K : ι → Set M) |
| (hK : ∀ i,IsCompact (K i)) (hKU : ∀ i,K i⊆(chartAt H (a i)).source) : |
| ∃ r D : ℝ,0 < r ∧ 1 ≤ D ∧ |
| ∀ (S : Diffeomorph I I M M ∞) (x : ℕ → M) (c e : ℕ → ι) |
| (P Q : ℕ → Matrix (Fin d) (Fin d) ℝ) (lam : Fin d → ℝ) (N : ℕ) (B : ℝ), |
| (∀ t≤N,x t∈cthickening r (K (c t))) → |
| (∀ t≤N,x t∈cthickening r (K (e t))) → |
| (∀ t<N,S (x t)=x (t+1)) → |
| (∀ t,P t*Q t=1 ∧ Q t*P t=1) → |
| (∀ t≤N,‖P t‖≤Real.exp B ∧ ‖Q t‖≤Real.exp B) → |
| (∀ t<N,(Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm |
| (coord I S (a (c t)) (a (c (t+1))) (x t))*P t= |
| P (t+1)*EuclideanSplitting.rate lam 1) → |
| ∃ P' Q' : ℕ → Matrix (Fin d) (Fin d) ℝ, |
| (∀ t≤N,P' t*Q' t=1 ∧ Q' t*P' t=1) ∧ |
| (∀ t≤N,‖P' t‖≤D*Real.exp B ∧ ‖Q' t‖≤D*Real.exp B) ∧ |
| ∀ t<N,(Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm |
| (coord I S (a (e t)) (a (e (t+1))) (x t))*P' t= |
| P' (t+1)*EuclideanSplitting.rate lam 1 := by |
| classical |
| obtain ⟨r,D,hr,hD,hh⟩ := uniform_chart_changes I a K hK hKU |
| refine ⟨r,D,hr,hD,fun S x c e P Q lam N B hc he hx hPQ hb hEq => ?_⟩ |
| let F := Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ) |
| have hmem t (ht : t≤N) := hh (c t) (e t) (x t) (hc t ht) (he t ht) |
| let R (t : ℕ) : Matrix (Fin d) (Fin d) ℝ := |
| F.symm (coord I (Diffeomorph.refl I M ∞) (a (c t)) (a (e t)) (x t)) |
| let V (t : ℕ) : Matrix (Fin d) (Fin d) ℝ := |
| F.symm (coord I (Diffeomorph.refl I M ∞) (a (e t)) (a (c t)) (x t)) |
| have hRV (t : ℕ) (ht : t≤N) : R t*V t=1 ∧ V t*R t=1 := by |
| have hm := hmem t ht |
| have hR := chartChange_coe I (a (c t)) (a (e t)) (x t) hm.1 hm.2.1 |
| have hV := chartChange_coe I (a (e t)) (a (c t)) (x t) hm.2.1 hm.1 |
| have hid : coord I (Diffeomorph.refl I M ∞) (a (c t)) (a (e t)) (x t) * |
| coord I (Diffeomorph.refl I M ∞) (a (e t)) (a (c t)) (x t) = 1 := by |
| have h := (chartChange I (a (c t)) (a (e t)) (x t) hm.1 hm.2.1).coe_comp_coe_symm |
| rw [chartChange_inverse I _ _ _ hm.1 hm.2.1] at h |
| simpa only [ContinuousLinearMap.mul_def, ContinuousLinearMap.one_def, hR, hV] using h |
| have hdi : coord I (Diffeomorph.refl I M ∞) (a (e t)) (a (c t)) (x t) * |
| coord I (Diffeomorph.refl I M ∞) (a (c t)) (a (e t)) (x t) = 1 := by |
| have h := (chartChange I (a (c t)) (a (e t)) (x t) hm.1 hm.2.1).coe_symm_comp_coe |
| rw [chartChange_inverse I _ _ _ hm.1 hm.2.1] at h |
| simpa only [ContinuousLinearMap.mul_def, ContinuousLinearMap.one_def, hR, hV] using h |
| constructor |
| · simpa only [map_mul,map_one,R,V] using congrArg F.symm hid |
| · simpa only [map_mul,map_one,R,V] using congrArg F.symm hdi |
| have hnorm (t : ℕ) (ht : t≤N) : ‖R t‖≤D ∧ ‖V t‖≤D := by |
| have hh' := (hmem t ht).2.2 |
| simpa only [R,V,←Matrix.l2_opNorm_toEuclideanCLM, |
| F,StarAlgEquiv.apply_symm_apply] using hh' |
| have hconj (t : ℕ) (ht : t<N) : F.symm (coord I S (a (e t)) (a (e (t+1))) (x t))= |
| R (t+1)*F.symm (coord I S (a (c t)) (a (c (t+1))) (x t))*V t := by |
| have h₀ := hmem t ht.le |
| have h₁ := hmem (t+1) (by omega) |
| have hd₀ : x t∈domain I S (a (c t)) (a (c (t+1))) := |
| ⟨h₀.1,by simpa only [Set.mem_preimage, hx t ht] using h₁.1⟩ |
| have hd₁ : x t∈domain I S (a (e t)) (a (e (t+1))) := |
| ⟨h₀.2.1,by simpa only [Set.mem_preimage, hx t ht] using h₁.2.1⟩ |
| rw [coord_chart_change I S _ _ _ _ _ hd₀ hd₁] |
| simp only [map_mul,chartChange_coe,hx t ht,R,V] |
| refine ⟨fun t => R t*P t,fun t => Q t*V t,fun t ht => ?_,fun t ht => ?_,fun t ht => ?_⟩ |
| · constructor |
| · calc |
| _=R t*(P t*Q t)*V t := by noncomm_ring |
| _=1 := by rw [(hPQ t).1,mul_one,(hRV t ht).1] |
| · calc |
| _=Q t*(V t*R t)*P t := by noncomm_ring |
| _=1 := by rw [(hRV t ht).2,mul_one,(hPQ t).2] |
| · constructor |
| · exact (norm_mul_le _ _).trans (mul_le_mul (hnorm t ht).1 (hb t ht).1 |
| (norm_nonneg _) (by linarith)) |
| · exact (norm_mul_le _ _).trans (by |
| calc |
| ‖Q t‖*‖V t‖ ≤ Real.exp B*D := mul_le_mul (hb t ht).2 (hnorm t ht).2 |
| (norm_nonneg _) (Real.exp_pos B).le |
| _=D*Real.exp B := mul_comm _ _) |
| · rw [hconj t ht] |
| calc |
| _=R (t+1)*F.symm (coord I S (a (c t)) (a (c (t+1))) (x t))*(V t*R t)*P t := by noncomm_ring |
| _=R (t+1)*(F.symm (coord I S (a (c t)) (a (c (t+1))) (x t))*P t) := by |
| rw [(hRV t ht.le).2,mul_one,mul_assoc] |
| _=(R (t+1)*P (t+1))*EuclideanSplitting.rate lam 1 := by rw [hEq t ht,mul_assoc] |
| end SmoothObstruction.ManifoldCoordinates |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Bundle Set Filter Topology Metric |
| open scoped Manifold ContDiff Matrix.Norms.L2Operator |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [MetricSpace M] [ProperSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
| {ι : Type*} [Fintype ι] |
|
|
| / |
| carried chart sequences lying in the enlarged finite atlas. All norm losses |
| are the same fixed D; the actual manifold-derivative conjugation identity |
| is proved above, and is not an extra cocycle assumption. -/ |
| theorem uniform_carried_gauge_explicit (a : ι → M) (K : ι → Set M) |
| (hK : ∀ i,IsCompact (K i)) (hKU : ∀ i,K i⊆(chartAt H (a i)).source) : |
| ∃ r D : ℝ,0 < r ∧ 1 ≤ D ∧ |
| ∀ (S : Diffeomorph I I M M ∞) (x : ℕ → M) (c e : ℕ → ι) |
| (P Q : ℕ → Matrix (Fin d) (Fin d) ℝ) (lam : Fin d → ℝ) (N : ℕ) (B : ℝ), |
| (∀ t≤N,x t∈cthickening r (K (c t))) → |
| (∀ t≤N,x t∈cthickening r (K (e t))) → |
| (∀ t<N,S (x t)=x (t+1)) → |
| (∀ t,P t*Q t=1 ∧ Q t*P t=1) → |
| (∀ t≤N,‖P t‖≤Real.exp B ∧ ‖Q t‖≤Real.exp B) → |
| (∀ t<N,(Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm |
| (coord I S (a (c t)) (a (c (t+1))) (x t))*P t= |
| P (t+1)*EuclideanSplitting.rate lam 1) → |
| ∃ P' Q' : ℕ → Matrix (Fin d) (Fin d) ℝ, |
| (∀ t≤N,P' t*Q' t=1 ∧ Q' t*P' t=1) ∧ |
| (∀ t≤N,‖P' t‖≤D*Real.exp B ∧ ‖Q' t‖≤D*Real.exp B) ∧ |
| (∀ t,P' t=(Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm |
| (coord I (Diffeomorph.refl I M ∞) (a (c t)) (a (e t)) (x t))*P t ∧ |
| Q' t=Q t*(Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm |
| (coord I (Diffeomorph.refl I M ∞) (a (e t)) (a (c t)) (x t))) ∧ |
| ∀ t<N,(Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm |
| (coord I S (a (e t)) (a (e (t+1))) (x t))*P' t= |
| P' (t+1)*EuclideanSplitting.rate lam 1 := by |
| classical |
| obtain ⟨r,D,hr,hD,hh⟩ := uniform_chart_changes I a K hK hKU |
| refine ⟨r,D,hr,hD,fun S x c e P Q lam N B hc he hx hPQ hb hEq => ?_⟩ |
| let F := Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ) |
| have hmem t (ht : t≤N) := hh (c t) (e t) (x t) (hc t ht) (he t ht) |
| let R (t : ℕ) : Matrix (Fin d) (Fin d) ℝ := |
| F.symm (coord I (Diffeomorph.refl I M ∞) (a (c t)) (a (e t)) (x t)) |
| let V (t : ℕ) : Matrix (Fin d) (Fin d) ℝ := |
| F.symm (coord I (Diffeomorph.refl I M ∞) (a (e t)) (a (c t)) (x t)) |
| have hRV (t : ℕ) (ht : t≤N) : R t*V t=1 ∧ V t*R t=1 := by |
| have hm := hmem t ht |
| have hR := chartChange_coe I (a (c t)) (a (e t)) (x t) hm.1 hm.2.1 |
| have hV := chartChange_coe I (a (e t)) (a (c t)) (x t) hm.2.1 hm.1 |
| have hid : coord I (Diffeomorph.refl I M ∞) (a (c t)) (a (e t)) (x t) * |
| coord I (Diffeomorph.refl I M ∞) (a (e t)) (a (c t)) (x t) = 1 := by |
| have h := (chartChange I (a (c t)) (a (e t)) (x t) hm.1 hm.2.1).coe_comp_coe_symm |
| rw [chartChange_inverse I _ _ _ hm.1 hm.2.1] at h |
| simpa only [ContinuousLinearMap.mul_def, ContinuousLinearMap.one_def, hR, hV] using h |
| have hdi : coord I (Diffeomorph.refl I M ∞) (a (e t)) (a (c t)) (x t) * |
| coord I (Diffeomorph.refl I M ∞) (a (c t)) (a (e t)) (x t) = 1 := by |
| have h := (chartChange I (a (c t)) (a (e t)) (x t) hm.1 hm.2.1).coe_symm_comp_coe |
| rw [chartChange_inverse I _ _ _ hm.1 hm.2.1] at h |
| simpa only [ContinuousLinearMap.mul_def, ContinuousLinearMap.one_def, hR, hV] using h |
| constructor |
| · simpa only [map_mul,map_one,R,V] using congrArg F.symm hid |
| · simpa only [map_mul,map_one,R,V] using congrArg F.symm hdi |
| have hnorm (t : ℕ) (ht : t≤N) : ‖R t‖≤D ∧ ‖V t‖≤D := by |
| have hh' := (hmem t ht).2.2 |
| simpa only [R,V,←Matrix.l2_opNorm_toEuclideanCLM, |
| F,StarAlgEquiv.apply_symm_apply] using hh' |
| have hconj (t : ℕ) (ht : t<N) : F.symm (coord I S (a (e t)) (a (e (t+1))) (x t))= |
| R (t+1)*F.symm (coord I S (a (c t)) (a (c (t+1))) (x t))*V t := by |
| have h₀ := hmem t ht.le |
| have h₁ := hmem (t+1) (by omega) |
| have hd₀ : x t∈domain I S (a (c t)) (a (c (t+1))) := |
| ⟨h₀.1,by simpa only [Set.mem_preimage, hx t ht] using h₁.1⟩ |
| have hd₁ : x t∈domain I S (a (e t)) (a (e (t+1))) := |
| ⟨h₀.2.1,by simpa only [Set.mem_preimage, hx t ht] using h₁.2.1⟩ |
| rw [coord_chart_change I S _ _ _ _ _ hd₀ hd₁] |
| simp only [map_mul,chartChange_coe,hx t ht,R,V] |
| refine ⟨fun t => R t*P t,fun t => Q t*V t,fun t ht => ?_,fun t ht => ?_,fun t => ⟨rfl,rfl⟩,fun t ht => ?_⟩ |
| · constructor |
| · calc |
| _=R t*(P t*Q t)*V t := by noncomm_ring |
| _=1 := by rw [(hPQ t).1,mul_one,(hRV t ht).1] |
| · calc |
| _=Q t*(V t*R t)*P t := by noncomm_ring |
| _=1 := by rw [(hRV t ht).2,mul_one,(hPQ t).2] |
| · constructor |
| · exact (norm_mul_le _ _).trans (mul_le_mul (hnorm t ht).1 (hb t ht).1 |
| (norm_nonneg _) (by linarith)) |
| · exact (norm_mul_le _ _).trans (by |
| calc |
| ‖Q t‖*‖V t‖ ≤ Real.exp B*D := mul_le_mul (hb t ht).2 (hnorm t ht).2 |
| (norm_nonneg _) (Real.exp_pos B).le |
| _=D*Real.exp B := mul_comm _ _) |
| · rw [hconj t ht] |
| calc |
| _=R (t+1)*F.symm (coord I S (a (c t)) (a (c (t+1))) (x t))*(V t*R t)*P t := by noncomm_ring |
| _=R (t+1)*(F.symm (coord I S (a (c t)) (a (c (t+1))) (x t))*P t) := by |
| rw [(hRV t ht.le).2,mul_one,mul_assoc] |
| _=(R (t+1)*P (t+1))*EuclideanSplitting.rate lam 1 := by rw [hEq t ht,mul_assoc] |
| end SmoothObstruction.ManifoldCoordinates |
|
|
| namespace SmoothObstruction.ShortBin |
| open scoped Matrix.Norms.L2Operator |
| structure Gauge (d L : ℕ) (K : ℝ) where |
| U : ℕ → Matrix (Fin d) (Fin d) ℝ |
| V : ℕ → Matrix (Fin d) (Fin d) ℝ |
| lam : Fin d → ℝ |
| UV : ∀ t,U t*V t=1 |
| VU : ∀ t,V t*U t=1 |
| normU : ∀ t≤L,‖U t‖≤Real.exp K |
| normV : ∀ t≤L,‖V t‖≤Real.exp K |
|
|
| end SmoothObstruction.ShortBin |
|
|
| namespace SmoothObstruction.ShortBin |
| open scoped Matrix.Norms.L2Operator |
|
|
| / |
| extension is algebraically invertible at every index and never affects an |
| actual derivative equation within the window. -/ |
| noncomputable def Gauge.ofFinite {d n : ℕ} {B : ℝ} |
| (P Q : ℕ → Matrix (Fin d) (Fin d) ℝ) (lam : Fin d → ℝ) |
| (hi : ∀ t≤n,P t*Q t=1 ∧ Q t*P t=1) |
| (hb : ∀ t≤n,‖P t‖≤Real.exp B ∧ ‖Q t‖≤Real.exp B) : Gauge d n B where |
| U t := P (min t n) |
| V t := Q (min t n) |
| lam := lam |
| UV t := (hi _ (Nat.min_le_right _ _)).1 |
| VU t := (hi _ (Nat.min_le_right _ _)).2 |
| normU t ht := by simpa only [Nat.min_eq_left ht] using (hb t ht).1 |
| normV t ht := by simpa only [Nat.min_eq_left ht] using (hb t ht).2 |
| end SmoothObstruction.ShortBin |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Bundle Set Filter Topology Metric MeasureTheory |
| open scoped Manifold ContDiff Matrix.Norms.L2Operator |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [MetricSpace M] [ProperSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
| {ι : Type*} [Fintype ι] |
|
|
| / |
| codes. Forward and backward gauges have exactly the SAME center bases, |
| not merely nearby center subspaces. A single uniform additive burden pays |
| all chart changes, independently of the number of iterates. -/ |
| theorem uniform_carried_bilateral (a : ι → M) (K : ι → Set M) |
| (hK : ∀ i,IsCompact (K i)) (hKU : ∀ i,K i⊆(chartAt H (a i)).source) : |
| ∃ r D : ℝ,0<r ∧ 1≤D ∧ |
| ∀ (S : Diffeomorph I I M M ∞) (c : M → ι) (x : M) (n : ℕ) (B R : ℝ) |
| (A : BilateralGauge I S a c n B R x) (eF eB : ℕ → ι), |
| (∀ y,y∈K (c y)) → eB 0=eF 0 → |
| (∀ t≤n,S^[t] x∈cthickening r (K (eF t))) → |
| (∀ t≤n,S.symm^[t] x∈cthickening r (K (eB t))) → |
| ∃ F Z : ShortBin.Gauge d n (B+Real.log D), |
| (∀ i,F.lam i=A.lam i) ∧ (∀ i,Z.lam i= -A.lam i) ∧ |
| Z.U 0=F.U 0 ∧ Z.V 0=F.V 0 ∧ |
| (∀ t<n,(Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm |
| (coord I S (a (eF t)) (a (eF (t+1))) (S^[t] x))*F.U t= |
| F.U (t+1)*EuclideanSplitting.rate F.lam 1) ∧ |
| ∀ t<n,(Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm |
| (coord I S.symm (a (eB t)) (a (eB (t+1))) (S.symm^[t] x))*Z.U t= |
| Z.U (t+1)*EuclideanSplitting.rate Z.lam 1 := by |
| obtain ⟨r,D,hr,hD,hcarry⟩ := uniform_carried_gauge_explicit I a K hK hKU |
| refine ⟨r,D,hr,hD,fun S c x n B R A eF eB hc h0 heF heB => ?_⟩ |
| have hDpos : 0<D := lt_of_lt_of_le zero_lt_one hD |
| have hbound : D*Real.exp B=Real.exp (B+Real.log D) := by |
| rw [Real.exp_add,Real.exp_log hDpos,mul_comm] |
| have hmF t (_ht : t≤n) : S^[t] x∈cthickening r (K (c (S^[t] x))) := |
| self_subset_cthickening _ (hc _) |
| have hmB t (_ht : t≤n) : S.symm^[t] x∈cthickening r (K (c (S.symm^[t] x))) := |
| self_subset_cthickening _ (hc _) |
| obtain ⟨PF,QF,hiF,hbF,formF,stepF⟩ := hcarry S (fun t => S^[t] x) |
| (fun t => c (S^[t] x)) eF A.UF A.VF A.lam n B hmF heF |
| (fun t _ => (Function.iterate_succ_apply' _ _ _).symm) A.invF A.normF A.stepF |
| obtain ⟨PB,QB,hiB,hbB,formB,stepB⟩ := hcarry S.symm (fun t => S.symm^[t] x) |
| (fun t => c (S.symm^[t] x)) eB A.UB A.VB (fun i => -A.lam i) n B hmB heB |
| (fun t _ => (Function.iterate_succ_apply' _ _ _).symm) A.invB A.normB A.stepB |
| have hbF' t (ht : t≤n) : ‖PF t‖≤Real.exp (B+Real.log D) ∧ ‖QF t‖≤Real.exp (B+Real.log D) := by |
| simpa only [hbound] using hbF t ht |
| have hbB' t (ht : t≤n) : ‖PB t‖≤Real.exp (B+Real.log D) ∧ ‖QB t‖≤Real.exp (B+Real.log D) := by |
| simpa only [hbound] using hbB t ht |
| refine ⟨ShortBin.Gauge.ofFinite PF QF A.lam hiF hbF', |
| ShortBin.Gauge.ofFinite PB QB (fun i => -A.lam i) hiB hbB', |
| fun _ => rfl,fun _ => rfl,?_,?_,?_,?_⟩ |
| · change PB (min 0 n)=PF (min 0 n) |
| rw [Nat.zero_min,(formB 0).1,(formF 0).1] |
| simp only [Function.iterate_zero_apply,h0,A.centerU] |
| · change QB (min 0 n)=QF (min 0 n) |
| rw [Nat.zero_min,(formB 0).2,(formF 0).2] |
| simp only [Function.iterate_zero_apply,h0,A.centerV] |
| · intro t ht |
| simpa only [ShortBin.Gauge.ofFinite,Nat.min_eq_left ht.le, |
| Nat.min_eq_left (show t+1≤n by omega)] using stepF t ht |
| · intro t ht |
| simpa only [ShortBin.Gauge.ofFinite,Nat.min_eq_left ht.le, |
| Nat.min_eq_left (show t+1≤n by omega)] using stepB t ht |
|
|
| theorem uniform_carried_bilateral_explicit (a : ι → M) (K : ι → Set M) |
| (hK : ∀ i,IsCompact (K i)) (hKU : ∀ i,K i⊆(chartAt H (a i)).source) : |
| ∃ r D : ℝ,0<r ∧ 1≤D ∧ |
| ∀ (S : Diffeomorph I I M M ∞) (c : M → ι) (x : M) (n : ℕ) (B R : ℝ) |
| (A : BilateralGauge I S a c n B R x) (eF eB : ℕ → ι), |
| (∀ y,y∈K (c y)) → eB 0=eF 0 → |
| (∀ t≤n,S^[t] x∈cthickening r (K (eF t))) → |
| (∀ t≤n,S.symm^[t] x∈cthickening r (K (eB t))) → |
| ∃ F Z : ShortBin.Gauge d n (B+Real.log D), |
| (∀ i,F.lam i=A.lam i) ∧ (∀ i,Z.lam i= -A.lam i) ∧ |
| Z.U 0=F.U 0 ∧ Z.V 0=F.V 0 ∧ |
| (∀ t<n,(Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm |
| (coord I S (a (eF t)) (a (eF (t+1))) (S^[t] x))*F.U t= |
| F.U (t+1)*EuclideanSplitting.rate F.lam 1) ∧ |
| (∀ t<n,(Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm |
| (coord I S.symm (a (eB t)) (a (eB (t+1))) (S.symm^[t] x))*Z.U t= |
| Z.U (t+1)*EuclideanSplitting.rate Z.lam 1) ∧ |
| ∀ t≤n, |
| F.U t=(Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm |
| (coord I (Diffeomorph.refl I M ∞) (a (c (S^[t] x))) (a (eF t)) (S^[t] x))*A.UF t ∧ |
| F.V t=A.VF t*(Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm |
| (coord I (Diffeomorph.refl I M ∞) (a (eF t)) (a (c (S^[t] x))) (S^[t] x)) ∧ |
| Z.U t=(Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm |
| (coord I (Diffeomorph.refl I M ∞) (a (c (S.symm^[t] x))) (a (eB t)) (S.symm^[t] x))*A.UB t ∧ |
| Z.V t=A.VB t*(Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm |
| (coord I (Diffeomorph.refl I M ∞) (a (eB t)) (a (c (S.symm^[t] x))) (S.symm^[t] x)) := by |
| obtain ⟨r,D,hr,hD,hcarry⟩ := uniform_carried_gauge_explicit I a K hK hKU |
| refine ⟨r,D,hr,hD,fun S c x n B R A eF eB hc h0 heF heB => ?_⟩ |
| have hDpos : 0<D := lt_of_lt_of_le zero_lt_one hD |
| have hbound : D*Real.exp B=Real.exp (B+Real.log D) := by |
| rw [Real.exp_add,Real.exp_log hDpos,mul_comm] |
| have hmF t (_ht : t≤n) : S^[t] x∈cthickening r (K (c (S^[t] x))) := |
| self_subset_cthickening _ (hc _) |
| have hmB t (_ht : t≤n) : S.symm^[t] x∈cthickening r (K (c (S.symm^[t] x))) := |
| self_subset_cthickening _ (hc _) |
| obtain ⟨PF,QF,hiF,hbF,formF,stepF⟩ := hcarry S (fun t => S^[t] x) |
| (fun t => c (S^[t] x)) eF A.UF A.VF A.lam n B hmF heF |
| (fun t _ => (Function.iterate_succ_apply' _ _ _).symm) A.invF A.normF A.stepF |
| obtain ⟨PB,QB,hiB,hbB,formB,stepB⟩ := hcarry S.symm (fun t => S.symm^[t] x) |
| (fun t => c (S.symm^[t] x)) eB A.UB A.VB (fun i => -A.lam i) n B hmB heB |
| (fun t _ => (Function.iterate_succ_apply' _ _ _).symm) A.invB A.normB A.stepB |
| have hbF' t (ht : t≤n) : ‖PF t‖≤Real.exp (B+Real.log D) ∧ ‖QF t‖≤Real.exp (B+Real.log D) := by |
| simpa only [hbound] using hbF t ht |
| have hbB' t (ht : t≤n) : ‖PB t‖≤Real.exp (B+Real.log D) ∧ ‖QB t‖≤Real.exp (B+Real.log D) := by |
| simpa only [hbound] using hbB t ht |
| refine ⟨ShortBin.Gauge.ofFinite PF QF A.lam hiF hbF', |
| ShortBin.Gauge.ofFinite PB QB (fun i => -A.lam i) hiB hbB', |
| fun _ => rfl,fun _ => rfl,?_,?_,?_,?_,?_⟩ |
| · change PB (min 0 n)=PF (min 0 n) |
| rw [Nat.zero_min,(formB 0).1,(formF 0).1] |
| simp only [Function.iterate_zero_apply,h0,A.centerU] |
| · change QB (min 0 n)=QF (min 0 n) |
| rw [Nat.zero_min,(formB 0).2,(formF 0).2] |
| simp only [Function.iterate_zero_apply,h0,A.centerV] |
| · intro t ht |
| simpa only [ShortBin.Gauge.ofFinite,Nat.min_eq_left ht.le, |
| Nat.min_eq_left (show t+1≤n by omega)] using stepF t ht |
| · intro t ht |
| simpa only [ShortBin.Gauge.ofFinite,Nat.min_eq_left ht.le, |
| Nat.min_eq_left (show t+1≤n by omega)] using stepB t ht |
| · intro t ht |
| simpa only [ShortBin.Gauge.ofFinite,Nat.min_eq_left ht] using |
| And.intro (formF t).1 (And.intro (formF t).2 (formB t)) |
|
|
| end SmoothObstruction.ManifoldCoordinates |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Bundle Set Filter Topology |
| open scoped Manifold ContDiff |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
|
|
| noncomputable def localMap (S : Diffeomorph I I M M ∞) (a b : M) : E (d:=d) → E (d:=d) := |
| extChartAt I b ∘ S ∘ (extChartAt I a).symm |
|
|
| / |
| coordinate map. Charts may be independently selected at each endpoint; |
| the formula is valid on half-space models without changing derivatives. -/ |
| theorem coord_eq_fderivWithin (S : Diffeomorph I I M M ∞) (a b x : M) |
| (hx : x∈domain I S a b) : |
| fderivWithin ℝ (localMap I S a b) (range I) ((extChartAt I a) x)=coord I S a b x := by |
| have hit : (extChartAt I a) x∈(extChartAt I a).target := |
| (extChartAt I a).map_source (by simpa only [extChartAt_source] using hx.1) |
| have hir : (extChartAt I a) x∈range I := extChartAt_target_subset_range a hit |
| have huniq : UniqueMDiffWithinAt (𝓘(ℝ,E (d:=d))) (range I) ((extChartAt I a) x) := |
| (I.uniqueDiffOn _ hir).uniqueMDiffWithinAt |
| have hs := mdifferentiableWithinAt_extChartAt_symm (I:=I) hit |
| have hf : MDifferentiableAt I (𝓘(ℝ,E (d:=d))) (extChartAt I b ∘ S) x := |
| (mdifferentiableAt_extChartAt hx.2).comp x (S.contMDiff.mdifferentiable (by simp) x) |
| have hcomp := mfderiv_comp_mfderivWithin_of_eq hf hs huniq |
| ((extChartAt I a).left_inv (by simpa only [extChartAt_source] using hx.1)) |
| have hchain := mfderiv_comp x (mdifferentiableAt_extChartAt (I:=I) hx.2) |
| (S.contMDiff.mdifferentiable (by simp) x) |
| rw [hchain] at hcomp |
| have hcoord := inTangentCoordinates_eq_mfderiv_comp |
| (I:=I) (I':=I) (f:=fun p : M×M => p.1) (g:=fun p : M×M => p.2) |
| (ϕ:=fun p => mfderiv I I S p.1) (x₀:=(a,b)) (x:=(x,S x)) hx.1 hx.2 |
| change coord I S a b x=_ at hcoord |
| rw [mfderivWithin_eq_fderivWithin] at hcomp |
| have he := hcomp.trans hcoord.symm |
| change fderivWithin ℝ ((extChartAt I b ∘ S) ∘ (extChartAt I a).symm) |
| (range I) ((extChartAt I a) x)=coord I S a b x at he |
| simpa only [localMap,Function.comp_assoc] using he |
| |
| /-- In an interior orbit tube this is an ordinary derivative, exactly the |
| input of the finite-array nonlinear construction. -/ |
| theorem hasFDerivAt_localMap (S : Diffeomorph I I M M ∞) (a b x : M) |
| (hx : x∈domain I S a b) |
| (hint : (extChartAt I a) x∈interior (range I)) : |
| HasFDerivAt (localMap I S a b) (coord I S a b x) ((extChartAt I a) x) := by |
| have hs := mdifferentiableWithinAt_extChartAt_symm (I:=I) |
| ((extChartAt I a).map_source (by simpa only [extChartAt_source] using hx.1)) |
| have hf : MDifferentiableAt I (𝓘(ℝ,E (d:=d))) (extChartAt I b ∘ S) x := |
| (mdifferentiableAt_extChartAt hx.2).comp x (S.contMDiff.mdifferentiable (by simp) x) |
| have hh := hf.comp_mdifferentiableWithinAt_of_eq _ hs ((extChartAt I a).left_inv (by simpa only [extChartAt_source] using hx.1)) |
| have hd : DifferentiableWithinAt ℝ (localMap I S a b) (range I) ((extChartAt I a) x) := by |
| exact mdifferentiableWithinAt_iff_differentiableWithinAt.mp hh |
| rw [←coord_eq_fderivWithin I S a b x hx] |
| exact hd.hasFDerivWithinAt.hasFDerivAt (mem_interior_iff_mem_nhds.mp hint) |
| end SmoothObstruction.ManifoldCoordinates |
| |
| namespace SmoothObstruction.ManifoldCoordinates |
| open Bundle Set Filter Topology |
| open scoped Manifold ContDiff |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
| |
| /-- Actual coordinate expressions are genuinely smooth in every retained |
| interior tube. For a boundary model no smooth extension is assumed here: |
| the statement uses the interior of the original model range. -/ |
| theorem contDiffAt_localMap (S : Diffeomorph I I M M ∞) (a b x : M) |
| (hx : x∈domain I S a b) |
| (hint : (extChartAt I a) x∈interior (range I)) : |
| ContDiffAt ℝ ∞ (localMap I S a b) ((extChartAt I a) x) := by |
| have hs := contMDiffWithinAt_extChartAt_symm_range (I:=I) (n:=∞) a |
| ((extChartAt I a).map_source (by simpa only [extChartAt_source] using hx.1)) |
| have hf : ContMDiffAt I (𝓘(ℝ,E (d:=d))) ∞ (extChartAt I b ∘ S) x := |
| (contMDiffAt_extChartAt' (I:=I) hx.2).comp x (S.contMDiff x) |
| have hh := hf.comp_contMDiffWithinAt_of_eq hs |
| ((extChartAt I a).left_inv (by simpa only [extChartAt_source] using hx.1)) |
| have hd : ContDiffWithinAt ℝ ∞ (localMap I S a b) (range I) |
| ((extChartAt I a) x) := contMDiffWithinAt_iff_contDiffWithinAt.mp hh |
| exact hd.contDiffAt (mem_interior_iff_mem_nhds.mp hint) |
|
|
| / |
| and have the expected derivative: this builds the endpoint maps used by the |
| inverse-function branch, not a new abstract replacement for DS^n. -/ |
| theorem projected_localMap {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] |
| (P : E (d:=d) →L[ℝ] F) (S : Diffeomorph I I M M ∞) (a b x : M) |
| (hx : x∈domain I S a b) |
| (hint : (extChartAt I a) x∈interior (range I)) : |
| ContDiffAt ℝ ∞ (P ∘ localMap I S a b) ((extChartAt I a) x) ∧ |
| HasFDerivAt (P ∘ localMap I S a b) (P.comp (coord I S a b x)) |
| ((extChartAt I a) x) := by |
| exact ⟨P.contDiff.contDiffAt.comp _ (contDiffAt_localMap I S a b x hx hint), |
| P.hasFDerivAt.comp _ (hasFDerivAt_localMap I S a b x hx hint)⟩ |
| end SmoothObstruction.ManifoldCoordinates |
|
|
| namespace SmoothObstruction.CarriedCharts |
| open Set Metric Filter Topology |
| open scoped ContDiff |
| variable {E F ι : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [ProperSpace E] [NormedAddCommGroup F] [NormedSpace ℝ F] [Fintype ι] |
|
|
| / |
| assumption. Ordinary derivatives are taken only in the original interior; |
| the bound is independent of distance to the boundary. -/ |
| theorem uniform_c2_boundary (f : ι → E → F) (K U : ι → Set E) (H : Set E) |
| (hK : ∀ i,IsCompact (K i)) (hU : ∀ i,IsOpen (U i)) (hH : IsClosed H) |
| (hKU : ∀ i,K i⊆U i) (huniq : ∀ i,UniqueDiffOn ℝ (U i∩H)) |
| (hf : ∀ i,ContDiffOn ℝ ∞ (f i) (U i∩H)) : |
| ∃ r C : ℝ,0 < r ∧ 1 ≤ C ∧ ∀ i x,x∈K i → |
| closedBall x r⊆U i ∧ ∀ y∈closedBall x r,y∈interior H → |
| ContDiffAt ℝ ∞ (f i) y ∧ |
| ‖fderiv ℝ (f i) y‖ ≤ C ∧ ‖fderiv ℝ (fderiv ℝ (f i)) y‖ ≤ C := by |
| have hex (i : ι) : ∃ r : ℝ,0 < r ∧ cthickening r (K i)⊆U i := |
| (hK i).exists_cthickening_subset_open (hU i) (hKU i) |
| choose r hr hrU using hex |
| obtain ⟨a,ha,har⟩ := finite_positive_lower r hr |
| let s (i : ι) := U i∩H |
| let f₁ (i : ι) := fderivWithin ℝ (f i) (s i) |
| let f₂ (i : ι) := fderivWithin ℝ (f₁ i) (s i) |
| have hf₁ (i : ι) : ContDiffOn ℝ ∞ (f₁ i) (s i) := |
| (hf i).fderivWithin (huniq i) (by simp) |
| have hf₂ (i : ι) : ContDiffOn ℝ ∞ (f₂ i) (s i) := |
| (hf₁ i).fderivWithin (huniq i) (by simp) |
| have hb (i : ι) : ∃ C : ℝ,∀ y∈cthickening (r i) (K i)∩H, |
| ‖f₁ i y‖ ≤ C ∧ ‖f₂ i y‖ ≤ C := by |
| have hcomp : IsCompact (cthickening (r i) (K i)∩H) := (hK i).cthickening.inter_right hH |
| have hsub : cthickening (r i) (K i)∩H⊆s i := fun y hy => ⟨hrU i hy.1,hy.2⟩ |
| obtain ⟨A,hA⟩ := hcomp.exists_bound_of_continuousOn (f := f₁ i) |
| ((hf₁ i).continuousOn.mono hsub) |
| obtain ⟨B,hB⟩ := hcomp.exists_bound_of_continuousOn (f := f₂ i) |
| ((hf₂ i).continuousOn.mono hsub) |
| exact ⟨max A B,fun y hy => ⟨(hA y hy).trans (le_max_left _ _), |
| (hB y hy).trans (le_max_right _ _)⟩⟩ |
| choose C hC using hb |
| let D : ℝ := 1+∑ i,|C i| |
| have hD : 1 ≤ D := by |
| dsimp [D] |
| have hh : 0≤∑ i,|C i| := Finset.sum_nonneg (fun i _ => abs_nonneg (C i)) |
| linarith |
| have hCD (i : ι) : C i ≤ D := by |
| have hh : |C i| ≤ ∑ j,|C j| := Finset.single_le_sum (fun j _ => abs_nonneg (C j)) |
| (Finset.mem_univ i) |
| dsimp [D] |
| linarith [le_abs_self (C i)] |
| refine ⟨a,D,ha,hD,fun i x hx => ?_⟩ |
| have hball : closedBall x a⊆cthickening (r i) (K i) := |
| (closedBall_subset_closedBall (har i)).trans (closedBall_subset_cthickening hx (r i)) |
| refine ⟨hball.trans (hrU i),fun y hy hyH => ?_⟩ |
| have hyU := hrU i (hball hy) |
| have hys : s i∈𝓝 y := inter_mem ((hU i).mem_nhds hyU) (mem_interior_iff_mem_nhds.mp hyH) |
| have heq : f₁ i =ᶠ[𝓝 y] fderiv ℝ (f i) := by |
| filter_upwards [((hU i).inter isOpen_interior).mem_nhds ⟨hyU,hyH⟩] with z hz |
| exact fderivWithin_of_mem_nhds (inter_mem ((hU i).mem_nhds hz.1) |
| (mem_interior_iff_mem_nhds.mp hz.2)) |
| have hfirst : f₁ i y=fderiv ℝ (f i) y := heq.eq_of_nhds |
| have hsecond : f₂ i y=fderiv ℝ (fderiv ℝ (f i)) y := |
| (fderivWithin_of_mem_nhds hys).trans heq.fderiv_eq |
| have hb' := hC i y ⟨hball hy,interior_subset hyH⟩ |
| rw [hfirst,hsecond] at hb' |
| exact ⟨(hf i y ⟨hyU,interior_subset hyH⟩).contDiffAt hys, |
| hb'.1.trans (hCD i),hb'.2.trans (hCD i)⟩ |
| end SmoothObstruction.CarriedCharts |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Set Metric Filter Topology |
| open scoped Manifold ContDiff |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
|
|
| / |
| coordinate domain. It is NOT a smooth extension of the expression. -/ |
| def domainU (S : Diffeomorph I I M M ∞) (a b : M) : Set (E (d:=d)) := |
| I.symm ⁻¹' ((chartAt H a).target∩(chartAt H a).symm ⁻¹' (S ⁻¹' (chartAt H b).source)) |
| |
| omit [IsManifold I ∞ M] in |
| lemma isOpen_domainU (S : Diffeomorph I I M M ∞) (a b : M) : IsOpen (domainU I S a b) := |
| ((chartAt H a).isOpen_inter_preimage_symm |
| ((chartAt H b).open_source.preimage S.continuous)).preimage I.continuous_symm |
| |
| omit [IsManifold I ∞ M] in |
| lemma map_mem_domainU (S : Diffeomorph I I M M ∞) (a b x : M) |
| (hx : x∈domain I S a b) : (extChartAt I a) x∈domainU I S a b := by |
| have ht := (extChartAt I a).map_source (by simpa only [extChartAt_source] using hx.1) |
| have htar : I.symm ((extChartAt I a) x)∈(chartAt H a).target := by |
| rw [extChartAt,OpenPartialHomeomorph.extend_target] at ht |
| exact ht.1 |
| refine ⟨htar,?_⟩ |
| change S ((extChartAt I a).symm ((extChartAt I a) x))∈(chartAt H b).source |
| rw [(extChartAt I a).left_inv (by simpa only [extChartAt_source] using hx.1)] |
| exact hx.2 |
| |
| theorem contDiffOn_localMap (S : Diffeomorph I I M M ∞) (a b : M) : |
| ContDiffOn ℝ ∞ (localMap I S a b) (domainU I S a b∩range I) := by |
| intro y hy |
| have hyt : y∈(extChartAt I a).target := by |
| rw [extChartAt,OpenPartialHomeomorph.extend_target] |
| exact ⟨hy.1.1,hy.2⟩ |
| have hby : S ((extChartAt I a).symm y)∈(chartAt H b).source := hy.1.2 |
| have hf : ContMDiffAt I (𝓘(ℝ,E (d:=d))) ∞ (extChartAt I b ∘ S) |
| ((extChartAt I a).symm y) := |
| (contMDiffAt_extChartAt' (I:=I) hby).comp _ (S.contMDiff _) |
| have hh := hf.comp_contMDiffWithinAt y |
| (contMDiffWithinAt_extChartAt_symm_range (I:=I) (n:=∞) a hyt) |
| have hd : ContDiffWithinAt ℝ ∞ (localMap I S a b) (range I) y := |
| contMDiffWithinAt_iff_contDiffWithinAt.mp hh |
| exact hd.mono inter_subset_right |
|
|
| / |
| used local chart pairs. Valid for the closed half-space model, uniformly |
| as the reference point approaches the boundary. No extension theorem or |
| long-iterate C2 bound has been assumed. -/ |
| theorem actual_uniform_c2 [T2Space M] {ι : Type*} [Fintype ι] |
| (hclosed : IsClosed (range I)) (S : Diffeomorph I I M M ∞) |
| (a : ι → M) (K : ι → Set M) (hK : ∀ i,IsCompact (K i)) |
| (hKU : ∀ i,K i⊆(chartAt H (a i)).source) : |
| ∃ r C : ℝ,0 < r ∧ 1 ≤ C ∧ ∀ i j x,x∈K i → S x∈K j → |
| closedBall ((extChartAt I (a i)) x) r⊆domainU I S (a i) (a j) ∧ |
| ∀ y∈closedBall ((extChartAt I (a i)) x) r,y∈interior (range I) → |
| ContDiffAt ℝ ∞ (localMap I S (a i) (a j)) y ∧ |
| ‖fderiv ℝ (localMap I S (a i) (a j)) y‖ ≤ C ∧ |
| ‖fderiv ℝ (fderiv ℝ (localMap I S (a i) (a j))) y‖ ≤ C := by |
| let L (ij : ι×ι) := (extChartAt I (a ij.1)) '' (K ij.1∩S ⁻¹' K ij.2) |
| have hcomp (ij : ι×ι) : IsCompact (L ij) := |
| ((hK ij.1).inter_right ((hK ij.2).isClosed.preimage S.continuous)).image_of_continuousOn |
| ((continuousOn_extChartAt (I:=I) (a ij.1)).mono (fun x hx => by simpa only [extChartAt_source] using hKU _ hx.1)) |
| have hsub (ij : ι×ι) : L ij⊆domainU I S (a ij.1) (a ij.2) := by |
| rintro z ⟨x,hx,rfl⟩ |
| exact map_mem_domainU I S (a ij.1) (a ij.2) x ⟨hKU _ hx.1,hKU _ hx.2⟩ |
| obtain ⟨r,C,hr,hC,hh⟩ := CarriedCharts.uniform_c2_boundary |
| (fun ij : ι×ι => localMap I S (a ij.1) (a ij.2)) L |
| (fun ij => domainU I S (a ij.1) (a ij.2)) (range I) hcomp |
| (fun ij => isOpen_domainU I S (a ij.1) (a ij.2)) hclosed hsub |
| (fun ij => by simpa only [inter_comm] using |
| I.uniqueDiffOn.inter (isOpen_domainU I S (a ij.1) (a ij.2))) |
| (fun ij => contDiffOn_localMap I S (a ij.1) (a ij.2)) |
| exact ⟨r,C,hr,hC,fun i j x hx hSx => hh (i,j) ((extChartAt I (a i)) x) ⟨x,⟨hx,hSx⟩,rfl⟩⟩ |
| end SmoothObstruction.ManifoldCoordinates |
| |
| namespace SmoothObstruction.ManifoldCoordinates |
| open Set Filter Topology |
| open scoped Manifold ContDiff |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
| |
| lemma interior_coord (a x : M) (hx : x∈(chartAt H a).source) : |
| (extChartAt I a) x∈interior (range I) ↔ I.IsInteriorPoint x := by |
| rw [I.isInteriorPoint_iff_of_mem_atlas (by simp : (∞ : ℕ∞ω)≠0) |
| (chart_mem_atlas H a) hx] |
| exact ⟨fun h => (chartAt H a).mem_interior_extend_target |
| ((chartAt H a).map_source hx) h, |
| fun h => (chartAt H a).interior_extend_target_subset_interior_range h⟩ |
| |
| /-- Carried local expressions preserve the *original* model interior on |
| exactly their actual relative domains, even for manifolds with boundary. -/ |
| theorem localMap_maps_interior (S : Diffeomorph I I M M ∞) (a b : M) : |
| MapsTo (localMap I S a b) (domainU I S a b∩interior (range I)) |
| (interior (range I)) := by |
| intro y hy |
| have hyt : y∈(extChartAt I a).target := by |
| rw [extChartAt,OpenPartialHomeomorph.extend_target] |
| exact ⟨hy.1.1,interior_subset hy.2⟩ |
| let x := (extChartAt I a).symm y |
| have hx : x∈(chartAt H a).source := by |
| simpa only [extChartAt_source] using (extChartAt I a).map_target hyt |
| have hxy : (extChartAt I a) x=y := (extChartAt I a).right_inv hyt |
| have hxi : I.IsInteriorPoint x := (interior_coord I a x hx).mp (hxy.symm ▸ hy.2) |
| have hSi : I.IsInteriorPoint (S x) := |
| ((S.isLocalDiffeomorph x).isInteriorPoint_iff (by simp)).mp hxi |
| have hSx : S x∈(chartAt H b).source := hy.1.2 |
| exact (interior_coord I b (S x) hSx).mpr hSi |
| |
| omit [IsManifold I ∞ M] in |
| lemma localMap_coordinates (S : Diffeomorph I I M M ∞) (a b x : M) |
| (hx : x∈(chartAt H a).source) : |
| localMap I S a b ((extChartAt I a) x)=(extChartAt I b) (S x) := by |
| simp only [localMap,Function.comp_apply, |
| (extChartAt I a).left_inv (by simpa only [extChartAt_source] using hx)] |
| end SmoothObstruction.ManifoldCoordinates |
| |
| namespace SmoothObstruction.StandardModels |
| open Set |
| open scoped Manifold |
| lemma real_closed (d : ℕ) : IsClosed (range (𝓡 d)) := by simp |
| lemma real_convex (d : ℕ) : Convex ℝ (range (𝓡 d)) := (𝓡 d).convex_range |
| lemma half_closed (d : ℕ) [NeZero d] : IsClosed (range (𝓡∂ d)) := by |
| rw [range_modelWithCornersEuclideanHalfSpace] |
| exact isClosed_le continuous_const (EuclideanSpace.proj (𝕜:=ℝ) (0:Fin d)).continuous |
| lemma half_convex (d : ℕ) [NeZero d] : Convex ℝ (range (𝓡∂ d)) := (𝓡∂ d).convex_range |
| end SmoothObstruction.StandardModels |
| |
| end |
| |
| end OAI |
| |
