| import OAI.Dynamics.EntropyCounterexample.C1Coordinates |
| import OAI.Dynamics.EntropyCounterexample.Conjugacy |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace Problem340 |
|
|
| theorem HasLocalC1Extension.comp |
| {X Y E F G : Type*} |
| [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] |
| [NormedAddCommGroup G] [NormedSpace ℝ G] |
| {eX : X → E} {eY : Y → F} {f : X → Y} {g : Y → G} |
| (hg : HasLocalC1Extension eY g) |
| (hf : HasLocalC1Extension eX (eY ∘ f)) : |
| HasLocalC1Extension eX (g ∘ f) := by |
| intro x |
| obtain ⟨U, hU, hxU, F, hF, hFe⟩ := hf x |
| simp only [Function.comp_apply] at hFe |
| obtain ⟨V, hV, hfxV, G, hG, hGe⟩ := hg (f x) |
| refine ⟨U ∩ F ⁻¹' V, hF.continuousOn.isOpen_inter_preimage hU hV, ?_, |
| G ∘ F, hG.comp_inter hF, ?_⟩ |
| · refine ⟨hxU, ?_⟩ |
| change F (eX x) ∈ V |
| rw [← hFe x hxU] |
| exact hfxV |
| · intro y hy |
| change g (f y) = G (F (eX y)) |
| rw [hGe (f y) (by rw [hFe y hy.1]; exact hy.2), hFe y hy.1] |
|
|
| theorem HasLocalC1Extension.conjugateMap |
| {X Y E F : Type*} [TopologicalSpace X] [TopologicalSpace Y] |
| [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] |
| {eX : X → E} {eY : Y → F} (e : X ≃ₜ Y) {f : X → X} |
| (hforward : HasLocalC1Extension eX (eY ∘ e)) |
| (hinverse : HasLocalC1Extension eY (eX ∘ e.symm)) |
| (hf : HasLocalC1Extension eX (eX ∘ f)) : |
| HasLocalC1Extension eY (eY ∘ conjugateMap e f) := |
| (hforward.comp hf).comp hinverse |
|
|
| theorem isC1SelfMap_conjugateMap_of_embeddings |
| {X E : Type*} [TopologicalSpace X] |
| [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {q : ℕ} {eX : X → E} (e : X ≃ₜ CounterexampleManifold q) {f : X → X} |
| (hforward : HasLocalC1Extension eX (standardEmbedding q ∘ e)) |
| (hinverse : HasLocalC1Extension (standardEmbedding q) (eX ∘ e.symm)) |
| (hf : HasLocalC1Extension eX (eX ∘ f)) : |
| IsC1SelfMap q (conjugateMap e f) := |
| isC1SelfMap_iff_hasLocalC1Extension.mpr |
| (HasLocalC1Extension.conjugateMap e hforward hinverse hf) |
|
|
| end Problem340 |
|
|
| end |
|
|
| end OAI |
|
|
