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a44267b | 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 | import OAI.NumberTheory.DukePrimeDegree.Dynamics.ShearingCollision
namespace OAI
/-!
# Fixed observable values at compact orbit limits
-/
noncomputable section
open Set Filter
open scoped Topology
namespace DukePrimeDegree
open PrimeDegreePackets
theorem compact_orbit_limit_with_data
{n : ℕ} {Z : Type*} [TopologicalSpace Z] [T2Space Z]
(K : Set (X n)) (hK : IsCompact K) (F : X n → Z) (hF : ContinuousOn F K)
(p : ℕ → X n) (H : ℕ → G n) (h : G n)
(hp : ∀ m, p m ∈ K) (hq : ∀ m, rightAction (p m) (H m) ∈ K)
(a b : Z) (ha : ∀ m, F (p m) = a) (hb : ∀ m, F (rightAction (p m) (H m)) = b)
(hH : Tendsto H atTop (𝓝 h)) :
∃ z ∈ K, rightAction z h ∈ K ∧ F z = a ∧ F (rightAction z h) = b := by
obtain ⟨z, hz, φ, hφ, hlim⟩ := hK.isSeqCompact hp
have hpLim : Tendsto (p ∘ φ) atTop (𝓝 z) := hlim
have hqLim : Tendsto (fun m => rightAction (p (φ m)) (H (φ m)))
atTop (𝓝 (rightAction z h)) :=
(continuous_rightAction_uncurry n).continuousAt.tendsto.comp
(hpLim.prodMk_nhds (hH.comp hφ.tendsto_atTop))
have hz' : rightAction z h ∈ K := hK.isClosed.mem_of_tendsto hqLim
(Eventually.of_forall fun m => hq (φ m))
have haLim : Tendsto (fun m => F (p (φ m))) atTop (𝓝 (F z)) :=
(hF z hz).tendsto.comp (tendsto_nhdsWithin_iff.mpr
⟨hpLim, Eventually.of_forall fun m => hp (φ m)⟩)
have hbLim : Tendsto (fun m => F (rightAction (p (φ m)) (H (φ m))))
atTop (𝓝 (F (rightAction z h))) :=
(hF _ hz').tendsto.comp (tendsto_nhdsWithin_iff.mpr
⟨hqLim, Eventually.of_forall fun m => hq (φ m)⟩)
refine ⟨z, hz, hz', ?_, ?_⟩
· have hc : Tendsto (fun m => F (p (φ m))) atTop (𝓝 a) := by
simpa only [ha] using (tendsto_const_nhds : Tendsto (fun _ : ℕ => a) atTop (𝓝 a))
exact tendsto_nhds_unique haLim hc
· have hc : Tendsto (fun m => F (rightAction (p (φ m)) (H (φ m)))) atTop (𝓝 b) := by
simpa only [hb] using (tendsto_const_nhds : Tendsto (fun _ : ℕ => b) atTop (𝓝 b))
exact tendsto_nhds_unique hbLim hc
end DukePrimeDegree
end
end OAI
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