| import OAI.Dynamics.EntropyCounterexample.CylinderModels |
| import OAI.Dynamics.EntropyCounterexample.EntropyLocalization |
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| namespace OAI |
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| noncomputable section |
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| open Filter Topology |
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| namespace Problem340 |
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| theorem entropy_zero_of_counterexample_orbit_data |
| (q : ℕ) (F : CounterexampleManifold q → CounterexampleManifold q) |
| (hF : Continuous F) |
| (clock₀ : StandardClock) (sphere₀ sphereInf : StandardSphere2) |
| (R : StandardSphere2 → StandardSphere2) (g : StandardClock → StandardClock) |
| (hzeroInv : Set.MapsTo F (clockMainCylinder clock₀ sphere₀ 0) |
| (clockMainCylinder clock₀ sphere₀ 0)) |
| (hInfInv : Set.MapsTo F |
| (sphereVectorCylinder (Function.update (fun _ : Fin (q + 1) => sphere₀) 0 sphereInf)) |
| (sphereVectorCylinder (Function.update (fun _ : Fin (q + 1) => sphere₀) 0 sphereInf))) |
| (hR : ∀ p ∈ clockMainCylinder clock₀ sphere₀ (0 : Fin (q + 1)), |
| ∀ k : Fin q, (F p).2 k.succ = R (p.2 k.succ)) |
| (hg : ∀ p ∈ sphereVectorCylinder |
| (Function.update (fun _ : Fin (q + 1) => sphere₀) 0 sphereInf), |
| (F p).1 = g p.1) |
| (hREntropy : Dynamics.coverEntropy |
| (fun v : Fin q → StandardSphere2 => fun k => R (v k)) Set.univ = 0) |
| (hgEntropy : Dynamics.coverEntropy g Set.univ = 0) |
| (horbit : ∀ x : CounterexampleManifold q, |
| (Tendsto (fun n => (F^[n] x).1) atTop (𝓝 clock₀) ∧ |
| Tendsto (fun n => (F^[n] x).2 0) atTop (𝓝 sphere₀)) ∨ |
| (Tendsto (fun n => (F^[n] x).2 0) atTop (𝓝 sphereInf) ∧ |
| ∀ k : Fin (q + 1), k ≠ 0 → ∀ᶠ n in atTop, (F^[n] x).2 k = sphere₀)) : |
| Dynamics.coverEntropy F (Set.univ : Set (CounterexampleManifold q)) = 0 := by |
| let vInf : Fin (q + 1) → StandardSphere2 := |
| Function.update (fun _ => sphere₀) 0 sphereInf |
| let K : Set (CounterexampleManifold q) := |
| clockMainCylinder clock₀ sphere₀ 0 ∪ sphereVectorCylinder vInf |
| have hK : IsCompact K := |
| counterexample_attractor_isCompact q clock₀ sphere₀ sphereInf 0 |
| have hKne : K.Nonempty := |
| (clockMainCylinder_nonempty clock₀ sphere₀ (0 : Fin (q + 1))).mono Set.subset_union_left |
| have hKInv : Set.MapsTo F K K := by |
| intro x hx |
| rcases hx with hx | hx |
| · exact Or.inl (hzeroInv hx) |
| · exact Or.inr (hInfInv hx) |
| have hKEntropy : Dynamics.coverEntropy F K = 0 := |
| coverEntropy_attractor_union_eq_zero q clock₀ sphere₀ vInf F R g |
| hzeroInv hInfInv hR hg hREntropy hgEntropy |
| apply coverEntropy_eq_zero_of_pointwise_attraction hF hK hKne hKInv hKEntropy |
| intro x |
| exact counterexample_attraction_of_coordinate_alternatives |
| (fun n => F^[n] x) clock₀ sphere₀ sphereInf 0 (horbit x) |
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| end Problem340 |
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| end |
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| end OAI |
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