File size: 3,756 Bytes
110f4f3 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 | import Mathlib
import OAI.Analysis.BiholderTransport.LocalFlow.MIntegralCurveUniformInvariant
namespace OAI
noncomputable section
open Set MeasureTheory Manifold Bundle
open scoped ContDiff Manifold ENNReal NNReal Topology
open Set Filter
open scoped Topology NNReal
open Set Filter
open scoped Topology
open Set Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set
open scoped Topology
open Set Filter Manifold Bundle ContinuousLinearMap
open scoped Topology ContDiff Manifold Bundle
open Set Filter ContinuousLinearMap InnerProductSpace
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
open scoped NNReal
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology
open MeasureTheory
open scoped ContDiff ENNReal
open Set Filter Manifold Bundle ContinuousLinearMap MeasureTheory
open scoped Topology ContDiff Manifold Bundle ENNReal
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology Manifold
open Set Filter Manifold Bundle ContinuousLinearMap
open scoped Topology ContDiff Manifold Bundle
open Set Filter Manifold Bundle
open scoped Topology ContDiff Manifold Bundle
open Set Filter Manifold Bundle
open scoped Topology ContDiff Manifold Bundle
open Set Filter Bundle
open scoped Topology Bundle
open scoped Topology
open Function Manifold Set
open Manifold Bundle
open scoped Manifold Bundle
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]
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))⟩
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
end WeakMTWTransport
end
end OAI
|