| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| namespace Problem340 | |
| def conjugateMap {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] | |
| (e : X ≃ₜ Y) (f : X → X) : Y → Y := fun y => e (f (e.symm y)) | |
| @[simp] theorem conjugateMap_apply {X Y : Type*} | |
| [TopologicalSpace X] [TopologicalSpace Y] (e : X ≃ₜ Y) (f : X → X) (x : X) : | |
| conjugateMap e f (e x) = e (f x) := by | |
| simp [conjugateMap] | |
| @[simp] theorem conjugateMap_symm {X Y : Type*} | |
| [TopologicalSpace X] [TopologicalSpace Y] (e : X ≃ₜ Y) (f : X → X) : | |
| conjugateMap e.symm (conjugateMap e f) = f := by | |
| funext x | |
| simp [conjugateMap] | |
| theorem continuous_conjugateMap {X Y : Type*} | |
| [TopologicalSpace X] [TopologicalSpace Y] (e : X ≃ₜ Y) | |
| {f : X → X} (hf : Continuous f) : Continuous (conjugateMap e f) := | |
| e.continuous.comp (hf.comp e.symm.continuous) | |
| theorem bijective_conjugateMap {X Y : Type*} | |
| [TopologicalSpace X] [TopologicalSpace Y] (e : X ≃ₜ Y) | |
| {f : X → X} (hf : Function.Bijective f) : Function.Bijective (conjugateMap e f) := | |
| e.bijective.comp (hf.comp e.symm.bijective) | |
| @[simp] theorem bijective_conjugateMap_iff {X Y : Type*} | |
| [TopologicalSpace X] [TopologicalSpace Y] (e : X ≃ₜ Y) (f : X → X) : | |
| Function.Bijective (conjugateMap e f) ↔ Function.Bijective f := by | |
| constructor | |
| · intro h | |
| simpa using bijective_conjugateMap e.symm h | |
| · exact bijective_conjugateMap e | |
| theorem coverEntropy_conjugateMap {X Y : Type*} | |
| [UniformSpace X] [UniformSpace Y] [CompactSpace X] [CompactSpace Y] | |
| (e : X ≃ₜ Y) (f : X → X) : | |
| Dynamics.coverEntropy (conjugateMap e f) (Set.univ : Set Y) = | |
| Dynamics.coverEntropy f (Set.univ : Set X) := by | |
| have h : Function.Semiconj e f (conjugateMap e f) := by | |
| intro x | |
| simp | |
| have h' : Function.Semiconj e.symm (conjugateMap e f) f := by | |
| intro y | |
| simp [conjugateMap] | |
| apply le_antisymm | |
| · simpa only [Set.image_univ, e.surjective.range_eq] using | |
| Dynamics.coverEntropy_image_le_of_uniformContinuous h | |
| (CompactSpace.uniformContinuous_of_continuous e.continuous) Set.univ | |
| · simpa only [Set.image_univ, e.symm.surjective.range_eq] using | |
| Dynamics.coverEntropy_image_le_of_uniformContinuous h' | |
| (CompactSpace.uniformContinuous_of_continuous e.symm.continuous) Set.univ | |
| end Problem340 | |
| end | |
| end OAI | |