| import OAI.Dynamics.ThreeTorus.SpectralTransport | |
| namespace OAI | |
| /-! Unconditional smooth simple Lebesgue spectrum theorem with real L2 cyclic generator. -/ | |
| section | |
| noncomputable section | |
| open Set Function Filter MeasureTheory | |
| open scoped Topology BigOperators | |
| namespace ThreeTorus | |
| /-- The unconditional main theorem. The generator and its | |
| entire basis are produced by the physical smooth diagonal, then transported by | |
| the fixed measure-preserving coordinate equivalence. -/ | |
| theorem simple_lebesgue_spectrum : MainConclusion := by | |
| obtain ⟨D,f,hf,hfr,ho,hd⟩ := exists_physical_basis | |
| let f₀ : physicalMeanZero := ⟨f,hf⟩ | |
| let v₀ (k : ℤ) : physicalMeanZero := ⟨D.orbit f k,D.orbit_centered hf k⟩ | |
| have hov : Orthonormal ℂ v₀ := by | |
| rw [orthonormal_iff_ite] | |
| intro k j | |
| exact orthonormal_iff_ite.mp ho k j | |
| have hdv : Dense (Submodule.span ℂ (Set.range v₀) : Set physicalMeanZero) := | |
| physical_subtype_dense v₀ hd | |
| let g := physicalZeroEquiv f₀ | |
| let v (k : ℤ) := physicalZeroEquiv (v₀ k) | |
| have hgr : IsReal g := by | |
| filter_upwards [physicalToTorus_ae f, | |
| measurePreserving_torusTriple.quasiMeasurePreserving.ae hfr] with x hx hy | |
| change (physicalToTorus f x).im=0 | |
| rw [hx] | |
| exact hy | |
| have hvg (k : ℤ) : ∀ᵐ x ∂μ, (v k).1 x=g.1 (((D.torusMap 1)^k) x) := by | |
| change ∀ᵐ x ∂μ, physicalToTorus (D.orbit f k) x=physicalToTorus f (((D.torusMap 1)^k) x) | |
| rw [D.torusMap_zpow] | |
| filter_upwards [physicalToTorus_ae (D.orbit f k), | |
| measurePreserving_torusTriple.quasiMeasurePreserving.ae (D.orbit_ae f k), | |
| (D.torusMap_mp k).quasiMeasurePreserving.ae (physicalToTorus_ae f)] with x hx hy hz | |
| simp only [D.torusMap_apply,torusTriple.apply_symm_apply] at hz | |
| exact hx.trans (hy.trans hz.symm) | |
| exact mainConclusion_of_basis (D.torusMap 1) (D.torusMap_smooth 1) | |
| (by rw [D.torusMap_symm]; exact D.torusMap_smooth (-1)) (D.torusMap_mp 1) | |
| g hgr v hvg (hov.comp_linearIsometryEquiv physicalZeroEquiv) | |
| (physicalZeroEquiv_dense_span v₀ hdv) | |
| end ThreeTorus | |
| end | |
| end | |
| end OAI | |