| import OAI.Analysis.IrrationalRotation.Rotation.CircleFiberDecay |
| import OAI.Analysis.IrrationalRotation.Analysis.FieldPowerBounds |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Filter MeasureTheory |
| open scoped Topology |
|
|
| namespace InvariantProjection.CircleFieldDecay |
|
|
| abbrev Hilbert := Lp Invariant233.WeightedShift.Hilbert 2 Invariant233.circleMeasure |
|
|
| theorem all_rates_fiber_decay {θ : ℝ} (hθ : Irrational θ) |
| {f : Invariant233.Circle → ℝ} (hf : Invariant233.IsContinuousCircleWeight f) |
| (w : Invariant233.Circle → Invariant233.WeightedShift.Weight) |
| (hw : ∀ x k, w x k = (f (x + k • (θ : Invariant233.Circle)) : ℂ)) : |
| ∀ a : ℝ, 0 < a → ∀ᶠ (n : ℕ) in atTop, ∀ x, |
| ‖Invariant233.WeightedShift.operator (w x) ^ n‖ ≤ Real.exp (-a * n) := by |
| intro a ha |
| filter_upwards [CircleWeightDecay.all_rates_decay_of_isContinuousCircleWeight hθ hf a ha] |
| with n hn |
| exact CircleFiberDecay.norm_fiber_power_le θ f (fun x => (hf.2.1 x).1) |
| w hw n (Real.exp_nonneg _) hn |
|
|
| theorem all_rates_norm_decay {θ : ℝ} (hθ : Irrational θ) |
| {f : Invariant233.Circle → ℝ} (hf : Invariant233.IsContinuousCircleWeight f) |
| (w : Invariant233.Circle → Invariant233.WeightedShift.Weight) |
| (hw : ∀ x k, w x k = (f (x + k • (θ : Invariant233.Circle)) : ℂ)) |
| (T : Hilbert →L[ℂ] Hilbert) |
| (hT : ∀ ξ : Hilbert, |
| ⇑(T ξ) =ᵐ[Invariant233.circleMeasure] |
| fun x => Invariant233.WeightedShift.operator (w x) (ξ x)) : |
| ∀ a : ℝ, 0 < a → ∀ᶠ (n : ℕ) in atTop, |
| ‖T ^ n‖ ≤ Real.exp (-a * n) := by |
| apply FieldPowerBounds.all_rates_norm_decay T |
| (fun x => Invariant233.WeightedShift.operator (w x)) hT |
| intro a ha |
| filter_upwards [all_rates_fiber_decay hθ hf w hw a ha] with n hn |
| exact Filter.Eventually.of_forall hn |
|
|
| theorem norm_quasinilpotent {θ : ℝ} (hθ : Irrational θ) |
| {f : Invariant233.Circle → ℝ} (hf : Invariant233.IsContinuousCircleWeight f) |
| (w : Invariant233.Circle → Invariant233.WeightedShift.Weight) |
| (hw : ∀ x k, w x k = (f (x + k • (θ : Invariant233.Circle)) : ℂ)) |
| (T : Hilbert →L[ℂ] Hilbert) |
| (hT : ∀ ξ : Hilbert, |
| ⇑(T ξ) =ᵐ[Invariant233.circleMeasure] |
| fun x => Invariant233.WeightedShift.operator (w x) (ξ x)) : |
| Invariant233.NormQuasinilpotent T := |
| Invariant233.norm_quasinilpotent_of_exponential_decay T |
| (all_rates_norm_decay hθ hf w hw T hT) |
|
|
| end InvariantProjection.CircleFieldDecay |
|
|
| end |
|
|
| end OAI |
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|
