| import Mathlib |
| import OAI.Geometry.SmoothYau.Spectrum.ContinuousEigenpairLinearization |
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| namespace OAI |
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| noncomputable section |
| open Set Filter Module |
| open scoped Topology |
| namespace YauCounterexamples |
| variable {X : Type*} [NormedAddCommGroup X] [NormedSpace ℝ X] [CompleteSpace X] |
| omit [CompleteSpace X] in |
| lemma compact_perturbed_eigenvectors (T : X →L[ℝ] X) (hT : IsCompactOperator T) |
| (S : ℕ → X →L[ℝ] X) (hS : Tendsto S atTop (𝓝 T)) {μ : ℝ} (hμ : μ ≠ 0) |
| (z : ℕ → X) (hz : ∀ n, ‖z n‖ = 1) (he : ∀ n, S n (z n) = μ • z n) : |
| ∃ v : X, ‖v‖ = 1 ∧ T v = μ • v ∧ |
| ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (z ∘ φ) atTop (𝓝 v) := by |
| obtain ⟨K,hK,hKi⟩ := hT.image_closedBall_subset_compact 1 |
| obtain ⟨y,_,φ,hφ,hy⟩ := hK.tendsto_subseq (fun n => |
| hKi ⟨z n, by simp only [Metric.mem_closedBall, dist_zero_right, hz n, le_refl], rfl⟩) |
| have hd : Tendsto (fun n => (S n - T) (z n)) atTop (𝓝 0) := by |
| apply tendsto_zero_iff_norm_tendsto_zero.mpr |
| have hb (n : ℕ) : ‖(S n - T) (z n)‖ ≤ ‖S n - T‖ := by |
| simpa only [hz, mul_one] using (S n - T).le_opNorm (z n) |
| have hl : Tendsto (fun n => ‖S n - T‖) atTop (𝓝 0) := by |
| simpa only [sub_self, norm_zero] using (hS.sub (tendsto_const_nhds (x := T))).norm |
| exact squeeze_zero (fun n => norm_nonneg _) hb hl |
| have hzlim : Tendsto (z ∘ φ) atTop (𝓝 (μ⁻¹ • y)) := by |
| have hh := (hy.add (hd.comp hφ.tendsto_atTop)).const_smul μ⁻¹ |
| simp only [add_zero] at hh |
| convert hh using 1 |
| funext n |
| change z (φ n) = μ⁻¹ • (T (z (φ n)) + (S (φ n) (z (φ n)) - T (z (φ n)))) |
| rw [he] |
| have hx : T (z (φ n)) + (μ • z (φ n) - T (z (φ n))) = μ • z (φ n) := by abel |
| rw [hx, smul_smul, inv_mul_cancel₀ hμ, one_smul] |
| have hn : ‖μ⁻¹ • y‖ = 1 := by |
| apply tendsto_nhds_unique hzlim.norm |
| simpa only [Function.comp_apply, hz] using (tendsto_const_nhds : |
| Tendsto (fun _ : ℕ => (1 : ℝ)) atTop (𝓝 1)) |
| have hte : T (μ⁻¹ • y) = μ • (μ⁻¹ • y) := by |
| have hc : Continuous (fun p : (X →L[ℝ] X) × X => p.1 p.2) := by fun_prop |
| have hh := hc.tendsto (T, μ⁻¹ • y) |>.comp |
| ((hS.comp hφ.tendsto_atTop).prodMk_nhds hzlim) |
| have hh' := hzlim.const_smul μ |
| apply tendsto_nhds_unique hh |
| simpa only [Function.comp_def, he] using hh' |
| exact ⟨_,hn,hte,φ,hφ,hzlim⟩ |
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|
| namespace PositivePairing |
| variable (B : PositivePairing (X := X)) |
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|
| omit [CompleteSpace X] in |
| theorem pinned_weak_first_variation (T : ℝ → X →L[ℝ] X) |
| (test : X → ℝ → X →L[ℝ] ℝ) (test₀ : X → X →L[ℝ] ℝ) |
| (hcompact : IsCompactOperator (T 0)) |
| (hT : Tendsto T (𝓝 (0 : ℝ)) (𝓝 (T 0))) |
| {μ : ℝ} (hμ : μ ≠ 0) {u : X} (hu0 : u ≠ 0) |
| (hu : ∀ᶠ t in 𝓝[≠] (0 : ℝ), T t u = μ • u) |
| (hlim : ∀ a, T 0 a = μ • a → |
| Tendsto (test a) (𝓝[≠] (0 : ℝ)) (𝓝 (test₀ a))) |
| (hvan : ∀ a, T 0 a = μ • a → ∀ᶠ t in 𝓝[≠] (0 : ℝ), |
| ∀ z, T t z = μ • z → test a t z = 0) |
| (hnd : ∀ z, T 0 z = μ • z → B.form u z = 0 → |
| (∀ v, T 0 v = μ • v → B.form u v = 0 → test₀ v z = 0) → z = 0) : |
| ∀ᶠ t in 𝓝[≠] (0 : ℝ), ∀ z, T t z = μ • z → ∃ c : ℝ, z = c • u := by |
| have hker : ∀ᶠ t in 𝓝[≠] (0 : ℝ), ∀ z, |
| T t z = μ • z → B.form u z = 0 → z = 0 := by |
| by_contra hbad |
| have hf : ∃ᶠ t in 𝓝[≠] (0 : ℝ), ∃ z, |
| T t z = μ • z ∧ B.form u z = 0 ∧ z ≠ 0 := by |
| simpa only [not_eventually, not_forall, Classical.not_imp, exists_prop] using hbad |
| obtain ⟨t,htlim,ht⟩ := exists_seq_forall_of_frequently hf |
| choose z hze hzp hzn using ht |
| let w : ℕ → X := fun n => ‖z n‖⁻¹ • z n |
| have hwn : ∀ n, ‖w n‖ = 1 := by |
| intro n |
| simp only [w, norm_smul, norm_inv, Real.norm_eq_abs, abs_norm, |
| inv_mul_cancel₀ (norm_ne_zero_iff.mpr (hzn n))] |
| have hwe : ∀ n, T (t n) (w n) = μ • w n := by |
| intro n |
| simp only [w, map_smul, hze] |
| exact smul_comm _ _ _ |
| have hwp : ∀ n, B.form u (w n) = 0 := by |
| intro n |
| simp only [w, map_smul, hzp, smul_zero] |
| have ht0 : Tendsto t atTop (𝓝 (0 : ℝ)) := htlim.mono_right nhdsWithin_le_nhds |
| obtain ⟨v,hvn,hve,φ,hφ,hv⟩ := compact_perturbed_eigenvectors (T 0) hcompact |
| (T ∘ t) (hT.comp ht0) hμ w hwn hwe |
| have hvp : B.form u v = 0 := by |
| apply tendsto_nhds_unique ((B.form u).continuous.tendsto v |>.comp hv) |
| simpa only [Function.comp_def, hwp] using (tendsto_const_nhds : |
| Tendsto (fun _ : ℕ => (0 : ℝ)) atTop (𝓝 0)) |
| have hzero : v = 0 := hnd v hve hvp (by |
| intro a hae _hap |
| have hc : Continuous (fun p : (X →L[ℝ] ℝ) × X => p.1 p.2) := by fun_prop |
| have hdv := hc.tendsto (test₀ a,v) |>.comp |
| (((hlim a hae).comp htlim |>.comp hφ.tendsto_atTop).prodMk_nhds hv) |
| apply tendsto_nhds_unique hdv |
| apply tendsto_const_nhds.congr' |
| filter_upwards [hφ.tendsto_atTop.eventually (htlim.eventually (hvan a hae))] with j hj |
| exact (hj _ (hwe (φ j))).symm) |
| rw [hzero, norm_zero] at hvn |
| exact zero_ne_one hvn |
| filter_upwards [hker,hu] with t ht htu z hz |
| let c : ℝ := B.form u z / B.form u u |
| have he : T t (z - c • u) = μ • (z - c • u) := by |
| simp only [map_sub, map_smul, htu, hz, smul_sub] |
| rw [smul_comm c μ u] |
| have hp : B.form u (z - c • u) = 0 := by |
| simp only [map_sub, map_smul, smul_eq_mul, c] |
| rw [div_mul_cancel₀ _ (B.positive u hu0).ne', sub_self] |
| exact ⟨c, sub_eq_zero.mp (ht _ he hp)⟩ |
| end PositivePairing |
| end YauCounterexamples |
| end |
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| end OAI |
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