| 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 |
| 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_uniform_time_on_compact {v : (x : M) → TangentSpace I x} |
| (hv : ContMDiff I (I.prod 𝓘(ℝ,E)) ∞ (fun x => (⟨x,v x⟩ : TangentBundle I M))) |
| {K : Set M} (hK : IsCompact K) : |
| ∃ ε : ℝ, 0 < ε ∧ ∀ x ∈ K, |
| ∃ γ : ℝ → M, γ 0 = x ∧ IsMIntegralCurveOn γ v (Ioo (-ε) ε) := by |
| classical |
| choose r hr U hU hflow using exists_uniform_local_manifold_curves hv |
| obtain ⟨t,_,ht⟩ := hK.elim_nhds_subcover U (fun x _ => hU x) |
| have hfin : ∀ t : Finset M, ∃ ε : ℝ, 0 < ε ∧ ∀ a ∈ t, ε ≤ r a := by |
| intro s |
| induction s using Finset.induction with |
| | empty => exact ⟨1,by norm_num,by simp⟩ |
| | @insert a s ha ih => |
| obtain ⟨ε,hε,hle⟩ := ih |
| refine ⟨min ε (r a),lt_min hε (hr a),?_⟩ |
| intro b hb |
| rcases Finset.mem_insert.mp hb with rfl | hb |
| · exact min_le_right _ _ |
| · exact (min_le_left _ _).trans (hle b hb) |
| obtain ⟨ε,hε,hle⟩ := hfin t |
| refine ⟨ε,hε,?_⟩ |
| intro x hx |
| obtain ⟨a,ha,hxa⟩ := mem_iUnion₂.mp (ht hx) |
| obtain ⟨γ,hγ0,hγ⟩ := hflow a x hxa |
| exact ⟨γ,hγ0,hγ.mono (Ioo_subset_Ioo (neg_le_neg (hle a ha)) (hle a ha))⟩ |
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| lemma exists_complete_curve_of_compact_invariant [T2Space M] |
| {v : (x : M) → TangentSpace I x} |
| (hv : ContMDiff I (I.prod 𝓘(ℝ,E)) ∞ (fun x => (⟨x,v x⟩ : TangentBundle I M))) |
| {K : Set M} (hK : IsCompact K) |
| (hinv : ∀ (a : ℝ) (γ : ℝ → M), γ 0 ∈ K → |
| IsMIntegralCurveOn γ v (Ioo (-a) a) → MapsTo γ (Ioo (-a) a) K) |
| (x : M) (hx : x ∈ K) : ∃ γ : ℝ → M, γ 0 = x ∧ IsMIntegralCurve γ v := by |
| obtain ⟨ε,hε,hflow⟩ := exists_uniform_time_on_compact hv hK |
| exact exists_isMIntegralCurve_of_uniform_invariant_set (hv.of_le (by simp)) |
| hε hflow hinv x hx |
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| end WeakMTWTransport |
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| end |
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| end OAI |
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