| import OAI.NumberTheory.DukePrimeDegree.Dynamics.ShearingCollision |
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|
| namespace OAI |
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| /-! |
| # Fixed observable values at compact orbit limits |
| -/ |
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|
| noncomputable section |
| open Set Filter |
| open scoped Topology |
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|
| namespace DukePrimeDegree |
| open PrimeDegreePackets |
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|
| 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 |
|
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| end DukePrimeDegree |
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| end |
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| end OAI |
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