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namespace OAI
noncomputable section
namespace Problem340
section Models
variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
def clockMainCylinderHomeomorph (q : ℕ) (x₀ : X) (y₀ : Y) :
↥(clockMainCylinder x₀ y₀ (0 : Fin (q + 1))) ≃ₜ (Fin q → Y) where
toFun p k := p.1.2 k.succ
invFun v := ⟨(x₀, Fin.cases y₀ v), rfl, rfl⟩
left_inv p := by
apply Subtype.ext
apply Prod.ext
· exact p.2.1.symm
· funext k
refine Fin.cases ?_ (fun _ => rfl) k
exact p.2.2.symm
right_inv v := rfl
continuous_toFun := by
apply continuous_pi
intro k
exact (continuous_apply k.succ).comp (continuous_snd.comp continuous_subtype_val)
continuous_invFun := by
apply Continuous.subtype_mk
apply Continuous.prodMk continuous_const
apply continuous_pi
intro k
refine Fin.cases ?_ (fun j => ?_) k
· exact continuous_const
· exact continuous_apply j
def sphereVectorCylinderHomeomorph {ι : Type*} (v₀ : ι → Y) :
↥(sphereVectorCylinder (X := X) v₀) ≃ₜ X where
toFun p := p.1.1
invFun x := ⟨(x, v₀), rfl⟩
left_inv p := by
apply Subtype.ext
exact Prod.ext rfl p.2.symm
right_inv x := rfl
continuous_toFun := continuous_fst.comp continuous_subtype_val
continuous_invFun := by
exact (continuous_id.prodMk continuous_const).subtype_mk _
theorem clockMainCylinder_semiconj (q : ℕ) (x₀ : X) (y₀ : Y)
(F : X × (Fin (q + 1) → Y) → X × (Fin (q + 1) → Y)) (R : Y → Y)
(hF : Set.MapsTo F (clockMainCylinder x₀ y₀ 0) (clockMainCylinder x₀ y₀ 0))
(hR : ∀ p ∈ clockMainCylinder x₀ y₀ (0 : Fin (q + 1)),
∀ k : Fin q, (F p).2 k.succ = R (p.2 k.succ)) :
Function.Semiconj (clockMainCylinderHomeomorph q x₀ y₀)
(hF.restrict F _ _) (fun v k => R (v k)) := by
intro p
funext k
exact hR p.1 p.2 k
theorem sphereVectorCylinder_semiconj {ι : Type*} (v₀ : ι → Y)
(F : X × (ι → Y) → X × (ι → Y)) (g : X → X)
(hF : Set.MapsTo F (sphereVectorCylinder v₀) (sphereVectorCylinder v₀))
(hg : ∀ p ∈ sphereVectorCylinder v₀, (F p).1 = g p.1) :
Function.Semiconj (sphereVectorCylinderHomeomorph (X := X) v₀)
(hF.restrict F _ _) g := by
intro p
exact hg p.1 p.2
end Models
end Problem340
namespace Problem340
theorem coverEntropy_univ_eq_of_compact_homeomorph
{X Y : Type*} [UniformSpace X] [UniformSpace Y]
[CompactSpace X] [CompactSpace Y]
(e : X ≃ₜ Y) (F : X → X) (G : Y → Y)
(h : Function.Semiconj e F G) :
Dynamics.coverEntropy F Set.univ = Dynamics.coverEntropy G Set.univ := by
have he : e '' Set.univ = Set.univ := Set.image_univ_of_surjective e.surjective
have hei : e.symm '' Set.univ = Set.univ := Set.image_univ_of_surjective e.symm.surjective
have hi : Function.Semiconj e.symm G F := by
intro y
apply e.injective
simpa using (h (e.symm y)).symm
apply le_antisymm
· simpa only [hei] using
Dynamics.coverEntropy_image_le_of_uniformContinuous hi
(CompactSpace.uniformContinuous_of_continuous e.symm.continuous) Set.univ
· simpa only [he] using
Dynamics.coverEntropy_image_le_of_uniformContinuous h
(CompactSpace.uniformContinuous_of_continuous e.continuous) Set.univ
theorem coverEntropy_clockMainCylinder
{X Y : Type*} [PseudoMetricSpace X] [PseudoMetricSpace Y]
[T2Space X] [T2Space Y] [CompactSpace X] [CompactSpace Y]
(q : ℕ) (x₀ : X) (y₀ : Y)
(F : X × (Fin (q + 1) → Y) → X × (Fin (q + 1) → Y)) (R : Y → Y)
(hF : Set.MapsTo F (clockMainCylinder x₀ y₀ 0) (clockMainCylinder x₀ y₀ 0))
(hR : ∀ p ∈ clockMainCylinder x₀ y₀ (0 : Fin (q + 1)),
∀ k : Fin q, (F p).2 k.succ = R (p.2 k.succ)) :
Dynamics.coverEntropy F (clockMainCylinder x₀ y₀ 0) =
Dynamics.coverEntropy (fun v : Fin q → Y => fun k => R (v k)) Set.univ := by
let : CompactSpace ↥(clockMainCylinder x₀ y₀ (0 : Fin (q + 1))) :=
isCompact_iff_compactSpace.mp (isClosed_clockMainCylinder x₀ y₀ 0).isCompact
rw [← Dynamics.coverEntropy_restrict hF]
exact coverEntropy_univ_eq_of_compact_homeomorph
(clockMainCylinderHomeomorph q x₀ y₀) _ _
(clockMainCylinder_semiconj q x₀ y₀ F R hF hR)
theorem coverEntropy_sphereVectorCylinder
{X Y ι : Type*} [PseudoMetricSpace X] [PseudoMetricSpace Y]
[T2Space X] [T2Space Y] [CompactSpace X] [CompactSpace Y] [Fintype ι]
(v₀ : ι → Y) (F : X × (ι → Y) → X × (ι → Y)) (g : X → X)
(hF : Set.MapsTo F (sphereVectorCylinder v₀) (sphereVectorCylinder v₀))
(hg : ∀ p ∈ sphereVectorCylinder v₀, (F p).1 = g p.1) :
Dynamics.coverEntropy F (sphereVectorCylinder v₀) =
Dynamics.coverEntropy g Set.univ := by
let : CompactSpace ↥(sphereVectorCylinder (X := X) v₀) :=
isCompact_iff_compactSpace.mp (isClosed_sphereVectorCylinder v₀).isCompact
rw [← Dynamics.coverEntropy_restrict hF]
exact coverEntropy_univ_eq_of_compact_homeomorph
(sphereVectorCylinderHomeomorph (X := X) v₀) _ _
(sphereVectorCylinder_semiconj v₀ F g hF hg)
theorem coverEntropy_attractor_union_eq_zero
{X Y : Type*} [PseudoMetricSpace X] [PseudoMetricSpace Y]
[T2Space X] [T2Space Y] [CompactSpace X] [CompactSpace Y]
(q : ℕ) (x₀ : X) (y₀ : Y) (v₀ : Fin (q + 1) → Y)
(F : X × (Fin (q + 1) → Y) → X × (Fin (q + 1) → Y))
(R : Y → Y) (g : X → X)
(hzero : Set.MapsTo F (clockMainCylinder x₀ y₀ 0) (clockMainCylinder x₀ y₀ 0))
(hinf : Set.MapsTo F (sphereVectorCylinder v₀) (sphereVectorCylinder v₀))
(hR : ∀ p ∈ clockMainCylinder x₀ y₀ (0 : Fin (q + 1)),
∀ k : Fin q, (F p).2 k.succ = R (p.2 k.succ))
(hg : ∀ p ∈ sphereVectorCylinder v₀, (F p).1 = g p.1)
(hREntropy : Dynamics.coverEntropy
(fun v : Fin q → Y => fun k => R (v k)) Set.univ = 0)
(hgEntropy : Dynamics.coverEntropy g Set.univ = 0) :
Dynamics.coverEntropy F
(clockMainCylinder x₀ y₀ 0 ∪ sphereVectorCylinder v₀) = 0 := by
rw [Dynamics.coverEntropy_union,
coverEntropy_clockMainCylinder q x₀ y₀ F R hzero hR,
coverEntropy_sphereVectorCylinder v₀ F g hinf hg,
hREntropy, hgEntropy, max_self]
end Problem340
end
end OAI
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