File size: 2,705 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 | import OAI.Dynamics.EntropyCounterexample.CylinderModels
import OAI.Dynamics.EntropyCounterexample.EntropyLocalization
namespace OAI
noncomputable section
open Filter Topology
namespace Problem340
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)
end Problem340
end
end OAI
|