| import Mathlib |
| import OAI.Analysis.BiholderTransport.LocalFlow.MIntegralCurveUniformInvariant |
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| namespace OAI |
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| noncomputable section |
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| open Set MeasureTheory Manifold Bundle |
| open scoped ContDiff Manifold ENNReal NNReal Topology |
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| open Set Filter |
| open scoped Topology NNReal |
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| open Set Filter |
| open scoped Topology |
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| open Set Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set |
| open scoped Topology |
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| open Set Filter Manifold Bundle ContinuousLinearMap |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter ContinuousLinearMap InnerProductSpace |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
| open scoped NNReal |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology |
| open MeasureTheory |
| open scoped ContDiff ENNReal |
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| open Set Filter Manifold Bundle ContinuousLinearMap MeasureTheory |
| open scoped Topology ContDiff Manifold Bundle ENNReal |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology Manifold |
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| open Set Filter Manifold Bundle ContinuousLinearMap |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Bundle |
| open scoped Topology Bundle |
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| open scoped Topology |
| open Function Manifold Set |
| open Manifold Bundle |
| open scoped Manifold Bundle |
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| namespace WeakMTWTransport |
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| section |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [CompleteSpace E] [HasContDiffBump E] |
| {H : Type*} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} |
| {M : Type*} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
| [I.Boundaryless] |
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| lemma exists_smooth_local_manifold_flow {v : (x : M) → TangentSpace I x} |
| (hv : ContMDiff I (I.prod 𝓘(ℝ,E)) ∞ |
| (fun x => (⟨x,v x⟩ : TangentBundle I M))) (a : M) : |
| ∃ r : ℝ, 0 < r ∧ ∃ U : Set M, IsOpen U ∧ a ∈ U ∧ |
| ∃ Φ : ℝ × M → M, |
| ContMDiffOn (𝓘(ℝ,ℝ).prod I) I ∞ Φ (Ioo (-r) r ×ˢ U) ∧ |
| (∀ x ∈ U, Φ (0,x) = x) ∧ |
| (∀ x ∈ U, IsMIntegralCurveOn (fun t => Φ (t,x)) v (Ioo (-r) r)) := by |
| let c := extChartAt I a |
| obtain ⟨r,hr,_,Ψ,hΨ,hΨ0,hΨder⟩ := exists_smooth_local_flow_on |
| (isOpen_extChartAt_target a) (contDiffOn_coordinate_vectorField hv a) |
| (c.map_source (mem_extChartAt_source a)) |
| let U := c.source ∩ c ⁻¹' Metric.ball (c a) r |
| have hU : IsOpen U := isOpen_extChartAt_preimage' a Metric.isOpen_ball |
| have ha : a ∈ U := ⟨mem_extChartAt_source a,Metric.mem_ball_self hr⟩ |
| let Φ : ℝ × M → M := fun z => c.symm (Ψ (z.1,c z.2)) |
| refine ⟨r,hr,U,hU,ha,Φ,?_,?_,?_⟩ |
| · intro z hz |
| have hc : ContMDiffAt I 𝓘(ℝ,E) ∞ c z.2 := |
| (contMDiffOn_extChartAt (I := I) (n := ∞) (x := a)).contMDiffAt |
| (by simpa only [extChartAt_source] using |
| (isOpen_extChartAt_source (I := I) a).mem_nhds hz.2.1) |
| have hg : ContDiffAt ℝ ∞ Ψ (z.1,c z.2) := |
| hΨ.contDiffAt ((isOpen_Ioo.prod Metric.isOpen_ball).mem_nhds ⟨hz.1,hz.2.2⟩) |
| have hsym : ContMDiffAt 𝓘(ℝ,E) I ∞ c.symm (Ψ (z.1,c z.2)) := |
| (contMDiffOn_extChartAt_symm a).contMDiffAt |
| ((isOpen_extChartAt_target a).mem_nhds (hΨder z.1 hz.1 (c z.2) hz.2.2).1) |
| have hpair : ContMDiffAt (𝓘(ℝ,ℝ).prod I) 𝓘(ℝ,ℝ × E) ∞ |
| (fun w : ℝ × M => (w.1,c w.2)) z := |
| contMDiffAt_prod_module_iff _ |>.mpr ⟨contMDiffAt_fst,hc.comp z contMDiffAt_snd⟩ |
| exact (hsym.comp z (hg.contMDiffAt.comp z hpair)).contMDiffWithinAt |
| · intro x hx |
| simp only [Φ,hΨ0 _ hx.2,c.left_inv hx.1] |
| · intro x hx t ht |
| exact (hasMFDerivAt_of_chart_ode a (hΨder t ht (c x) hx.2).1 |
| (hΨder t ht (c x) hx.2).2).hasMFDerivWithinAt |
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| end |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [CompleteSpace E] [HasContDiffBump E] |
| {H : Type*} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} |
| {M : Type*} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
| [I.Boundaryless] [T2Space M] |
| {v : (x : M) → TangentSpace I x} |
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| variable (hc : ∀ x : M, ∃ γ : ℝ → M, γ 0 = x ∧ IsMIntegralCurve γ v) |
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| omit [CompleteSpace E] [HasContDiffBump E] [IsManifold I ∞ M] [I.Boundaryless] [T2Space M] in |
| noncomputable def completeFlow (t : ℝ) (x : M) : M := Classical.choose (hc x) t |
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| omit [CompleteSpace E] [HasContDiffBump E] [IsManifold I ∞ M] [I.Boundaryless] [T2Space M] in |
| lemma completeFlow_zero (x : M) : completeFlow hc 0 x = x := (Classical.choose_spec (hc x)).1 |
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| omit [CompleteSpace E] [HasContDiffBump E] [IsManifold I ∞ M] [I.Boundaryless] [T2Space M] in |
| lemma completeFlow_curve (x : M) : IsMIntegralCurve (fun t => completeFlow hc t x) v := |
| (Classical.choose_spec (hc x)).2 |
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| variable (hv : ContMDiff I (I.prod 𝓘(ℝ,E)) ∞ (fun x => (⟨x,v x⟩ : TangentBundle I M))) |
| include hv |
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| omit [CompleteSpace E] [HasContDiffBump E] in |
| lemma completeFlow_add (s t : ℝ) (x : M) : |
| completeFlow hc (s+t) x = completeFlow hc s (completeFlow hc t x) := by |
| have he := isMIntegralCurve_Ioo_eq_of_contMDiff_boundaryless (t₀ := 0) |
| (hv.of_le (by simp)) ((completeFlow_curve hc x).comp_add t) |
| (completeFlow_curve hc (completeFlow hc t x)) |
| (by simpa only [Function.comp_apply,zero_add] using |
| (completeFlow_zero hc (completeFlow hc t x)).symm) |
| exact congrFun he s |
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| omit [CompleteSpace E] [HasContDiffBump E] in |
| lemma completeFlow_eq_local {r : ℝ} (hr : 0 < r) {U : Set M} {Φ : ℝ × M → M} |
| (hΦ0 : ∀ x ∈ U, Φ (0,x) = x) |
| (hΦode : ∀ x ∈ U, IsMIntegralCurveOn (fun t => Φ (t,x)) v (Ioo (-r) r)) |
| {x : M} (hx : x ∈ U) {t : ℝ} (ht : t ∈ Ioo (-r) r) : |
| completeFlow hc t x = Φ (t,x) := by |
| apply isMIntegralCurveOn_Ioo_eqOn_of_contMDiff_boundaryless |
| (t₀ := 0) ⟨neg_lt_zero.mpr hr,hr⟩ (hv.of_le (by simp)) |
| ((completeFlow_curve hc x).isMIntegralCurveOn _) (hΦode x hx) _ ht |
| exact (completeFlow_zero hc x).trans (hΦ0 x hx).symm |
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| omit [CompleteSpace E] [HasContDiffBump E] in |
| lemma completeFlow_smooth_step {r : ℝ} (hr : 0 < r) {U : Set M} (hU : IsOpen U) |
| {Φ : ℝ × M → M} |
| (hΦ : ContMDiffOn (𝓘(ℝ,ℝ).prod I) I ∞ Φ (Ioo (-r) r ×ˢ U)) |
| (hΦ0 : ∀ x ∈ U, Φ (0,x) = x) |
| (hΦode : ∀ x ∈ U, IsMIntegralCurveOn (fun t => Φ (t,x)) v (Ioo (-r) r)) |
| {s t : ℝ} {a : M} (hs : ContMDiffAt I I ∞ (completeFlow hc s) a) |
| (ha : completeFlow hc s a ∈ U) (ht : t-s ∈ Ioo (-r) r) : |
| ContMDiffAt I I ∞ (completeFlow hc t) a := by |
| have hsm := (hΦ.contMDiffAt ((isOpen_Ioo.prod hU).mem_nhds ⟨ht,ha⟩)).comp a |
| (contMDiffAt_const.prodMk hs) |
| apply hsm.congr_of_eventuallyEq |
| have hnear := hs.continuousAt.preimage_mem_nhds (hU.mem_nhds ha) |
| filter_upwards [hnear] with x hx |
| change completeFlow hc t x = Φ (t-s,completeFlow hc s x) |
| rw [← completeFlow_eq_local hc hv hr hΦ0 hΦode hx ht,← completeFlow_add hc hv,sub_add_cancel] |
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| lemma contMDiff_completeFlow_time_slice (t : ℝ) : ContMDiff I I ∞ (completeFlow hc t) := by |
| intro a |
| let T : Set ℝ := {s | ContMDiffAt I I ∞ (completeFlow hc s) a} |
| have hT0 : 0 ∈ T := by |
| change ContMDiffAt I I ∞ (completeFlow hc 0) a |
| have hzero : completeFlow hc 0 = id := funext (completeFlow_zero hc) |
| rw [hzero] |
| exact contMDiffAt_id |
| have hopen : IsOpen T := by |
| apply Metric.isOpen_iff.mpr |
| intro s hs |
| obtain ⟨r,hr,U,hU,hUa,Φ,hΦ,hΦ0,hΦode⟩ := |
| exists_smooth_local_manifold_flow hv (completeFlow hc s a) |
| refine ⟨r,hr,?_⟩ |
| intro t ht |
| exact completeFlow_smooth_step hc hv hr hU hΦ hΦ0 hΦode hs hUa |
| (by simpa only [Metric.mem_ball,Real.dist_eq,abs_lt,mem_Ioo] using ht) |
| have hcompl : IsOpen Tᶜ := by |
| apply Metric.isOpen_iff.mpr |
| intro t ht |
| obtain ⟨r,hr,U,hU,hUa,Φ,hΦ,hΦ0,hΦode⟩ := |
| exists_smooth_local_manifold_flow hv (completeFlow hc t a) |
| have hnear := (completeFlow_curve hc a).continuous.continuousAt.preimage_mem_nhds |
| (hU.mem_nhds hUa) |
| obtain ⟨δ,hδ,hδU⟩ := Metric.mem_nhds_iff.mp hnear |
| refine ⟨min r δ,lt_min hr hδ,?_⟩ |
| intro s hs hsm |
| have hsr : dist s t < r := lt_of_lt_of_le hs (min_le_left _ _) |
| have hsδ : dist s t < δ := lt_of_lt_of_le hs (min_le_right _ _) |
| apply ht |
| exact completeFlow_smooth_step hc hv hr hU hΦ hΦ0 hΦode hsm (hδU hsδ) |
| (by simpa only [Real.dist_eq,abs_sub_comm,abs_lt,mem_Ioo] using hsr) |
| have hTuniv : T = univ := (IsClopen.eq_univ ⟨isOpen_compl_iff.mp hcompl,hopen⟩ ⟨0,hT0⟩) |
| change t ∈ T |
| rw [hTuniv] |
| exact mem_univ t |
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| lemma contMDiff_completeFlow : |
| ContMDiff (𝓘(ℝ,ℝ).prod I) I ∞ (fun z : ℝ × M => completeFlow hc z.1 z.2) := by |
| intro z |
| obtain ⟨r,hr,U,hU,hUa,Φ,hΦ,hΦ0,hΦode⟩ := |
| exists_smooth_local_manifold_flow hv (completeFlow hc z.1 z.2) |
| have hs := (contMDiff_completeFlow_time_slice hc hv z.1) z.2 |
| have hpair : ContMDiffAt (𝓘(ℝ,ℝ).prod I) (𝓘(ℝ,ℝ).prod I) ∞ |
| (fun w : ℝ × M => (w.1-z.1,completeFlow hc z.1 w.2)) z := |
| (contMDiffAt_fst.sub contMDiffAt_const).prodMk (hs.comp z contMDiffAt_snd) |
| have hsm := (hΦ.contMDiffAt ((isOpen_Ioo.prod hU).mem_nhds |
| (show (z.1-z.1,completeFlow hc z.1 z.2) ∈ Ioo (-r) r ×ˢ U from |
| ⟨by simp only [sub_self,mem_Ioo]; exact ⟨neg_lt_zero.mpr hr,hr⟩,hUa⟩))).comp z hpair |
| apply hsm.congr_of_eventuallyEq |
| have hnear := hpair.continuousAt.preimage_mem_nhds |
| ((isOpen_Ioo.prod hU).mem_nhds |
| (show (z.1-z.1,completeFlow hc z.1 z.2) ∈ Ioo (-r) r ×ˢ U from |
| ⟨by simp only [sub_self,mem_Ioo]; exact ⟨neg_lt_zero.mpr hr,hr⟩,hUa⟩)) |
| filter_upwards [hnear] with w hw |
| change completeFlow hc w.1 w.2 = Φ (w.1-z.1,completeFlow hc z.1 w.2) |
| rw [← completeFlow_eq_local hc hv hr hΦ0 hΦode hw.2 hw.1, |
| ← completeFlow_add hc hv,sub_add_cancel] |
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| end WeakMTWTransport |
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| end |
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| end OAI |
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