| import OAI.Dynamics.EntropyCounterexample.PolynomialSpectralBound | |
| import OAI.Geometry.SingularHomology.HomologyTransfer | |
| import OAI.Dynamics.EntropyCounterexample.FiniteSpectrumBound | |
| namespace OAI | |
| noncomputable section | |
| namespace Problem340 | |
| open PolynomialSpectrum | |
| variable {X Y : Type} [TopologicalSpace X] [TopologicalSpace Y] | |
| theorem homology_pairRadii_factor_subset (n : ℕ) | |
| (f : X → X) (i : Y → X) (p : X → Y) | |
| (hf : Continuous f) (hi : Continuous i) (hp : Continuous p) | |
| (H : ContinuousMap.Homotopy (continuousMapOf f hf) | |
| (continuousMapOf (i ∘ p) (hi.comp hp))) : | |
| pairRadii (realHomologyMap n f hf) ⊆ | |
| insert 0 (pairRadii (realHomologyMap n (p ∘ i) (hp.comp hi))) := by | |
| rw [realHomologyMap_eq_of_homotopy n hf (hi.comp hp) H, | |
| realHomologyMap_comp n p i hp hi, realHomologyMap_comp n i p hi hp] | |
| exact pairRadii_comp_subset _ _ | |
| theorem bddAbove_homology_pairRadii_of_homotopy_factor | |
| (f : X → X) (i : Y → X) (p : X → Y) | |
| (hf : Continuous f) (hi : Continuous i) (hp : Continuous p) | |
| (H : ContinuousMap.Homotopy (continuousMapOf f hf) | |
| (continuousMapOf (i ∘ p) (hi.comp hp))) | |
| (hB : BddAbove (⋃ n, pairRadii (realHomologyMap n (p ∘ i) (hp.comp hi)))) : | |
| BddAbove (⋃ n, pairRadii (realHomologyMap n f hf)) := by | |
| apply (hB.insert 0).mono | |
| intro r hr | |
| obtain ⟨n, hn⟩ := Set.mem_iUnion.mp hr | |
| rcases homology_pairRadii_factor_subset n f i p hf hi hp H hn with h | h | |
| · exact Set.mem_insert_iff.mpr (Or.inl h) | |
| · exact Set.mem_insert_of_mem _ (Set.mem_iUnion.mpr ⟨n, h⟩) | |
| theorem bddAbove_homology_pairRadii_of_finite_homotopy_factor | |
| (f : X → X) (i : Y → X) (p : X → Y) | |
| (hf : Continuous f) (hi : Continuous i) (hp : Continuous p) | |
| (H : ContinuousMap.Homotopy (continuousMapOf f hf) | |
| (continuousMapOf (i ∘ p) (hi.comp hp))) | |
| [∀ n, FiniteDimensional ℝ (RealSingularHomology n Y)] | |
| (N : ℕ) (hvanish : ∀ n, N < n → Subsingleton (RealSingularHomology n Y)) : | |
| BddAbove (⋃ n, pairRadii (realHomologyMap n f hf)) := by | |
| apply bddAbove_homology_pairRadii_of_homotopy_factor f i p hf hi hp H | |
| have h := bddAbove_total_pairedEigenvalue_moduli | |
| (fun n => realHomologyMap n (p ∘ i) (hp.comp hi)) N hvanish | |
| simpa only [pairRadii, Set.iUnion_ofPred] using h | |
| theorem totalRealHomologySpectralRadius_eq_sSup_pairRadii | |
| (f : X → X) (hf : Continuous f) : | |
| totalRealHomologySpectralRadius f hf = | |
| sSup (⋃ n, pairRadii (realHomologyMap n f hf)) := by | |
| simp only [totalRealHomologySpectralRadius, pairRadii, Set.iUnion_ofPred] | |
| end Problem340 | |
| end | |
| end OAI | |