File size: 2,352 Bytes
92416a2 | 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 | 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
|