| import OAI.Geometry.SingularHomology.MapHomology |
| import OAI.Dynamics.EntropyCounterexample.RealizedConstruction |
| import OAI.Dynamics.EntropyCounterexample.HomotopySpectrum |
| import OAI.Dynamics.EntropyCounterexample.SpectralRadius |
|
|
| namespace OAI |
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| noncomputable section |
|
|
| namespace Problem340.Construction |
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|
| theorem continuous_realizedMap_actual (L : ℝ) (hL : 2 < L) : |
| Continuous (realizedMap L hL) := |
| continuous_realizedMap L hL (concreteMap_continuous_for_homotopy L hL) |
|
|
| theorem realizedMap_eigenvector_of_sphere (n : ℕ) (L : ℝ) (hL : 2 < L) |
| (c : ℝ) (v : RealSingularHomology n Sphere) (hv : v ≠ 0) |
| (heig : realHomologyMap n rawSphereSquare continuous_rawSphereSquare v = c • v) : |
| ∃ w : RealSingularHomology n (CounterexampleManifold 3), w ≠ 0 ∧ |
| realHomologyMap n (realizedMap L hL) (continuous_realizedMap_actual L hL) w = c • w := by |
| obtain ⟨w, hw, he⟩ := concreteMap_eigenvector_of_sphere n L hL c v hv heig |
| exact realizedMap_eigenvector L hL (concreteMap_continuous_for_homotopy L hL) |
| n c w hw he |
|
|
| theorem totalRealHomologySpectralRadius_realizedMap_eq_normalForm |
| (L : ℝ) (hL : 2 < L) : |
| totalRealHomologySpectralRadius (realizedMap L hL) |
| (continuous_realizedMap_actual L hL) = |
| totalRealHomologySpectralRadius (@rawNormalForm 3) continuous_rawNormalForm := by |
| rw [totalRealHomologySpectralRadius_realizedMap] |
| exact (totalRealHomologySpectralRadius_eq_of_homotopy continuous_rawNormalForm |
| (concreteMap_continuous_for_homotopy L hL) (concreteMapHomotopy L hL)).symm |
|
|
| theorem homologyEigenvalueRadii_concreteMap_eq_normalForm (L : ℝ) (hL : 2 < L) : |
| homologyEigenvalueRadii (concreteMap L hL) (concreteMap_continuous_for_homotopy L hL) = |
| homologyEigenvalueRadii (@rawNormalForm 3) continuous_rawNormalForm := |
| (homologyRadiusSet_eq_of_homotopy continuous_rawNormalForm |
| (concreteMap_continuous_for_homotopy L hL) (concreteMapHomotopy L hL)).symm |
|
|
| theorem log_two_le_realizedMap_radius_of_sphere (L : ℝ) (hL : 2 < L) |
| (v : RealSingularHomology 2 Sphere) (hv : v ≠ 0) |
| (heig : realHomologyMap 2 rawSphereSquare continuous_rawSphereSquare v = (2 : ℝ) • v) |
| (hbounded : BddAbove |
| (homologyEigenvalueRadii (@rawNormalForm 3) continuous_rawNormalForm)) : |
| Real.log (2 : ℝ) ≤ Real.log (totalRealHomologySpectralRadius |
| (realizedMap L hL) (continuous_realizedMap_actual L hL)) := by |
| obtain ⟨w, hw, he⟩ := concreteMap_eigenvector_of_sphere 2 L hL 2 v hv heig |
| have hb : BddAbove (homologyEigenvalueRadii (concreteMap L hL) |
| (concreteMap_continuous_for_homotopy L hL)) := by |
| rw [homologyEigenvalueRadii_concreteMap_eq_normalForm] |
| exact hbounded |
| rw [totalRealHomologySpectralRadius_realizedMap] |
| exact log_two_le_log_totalRealHomologySpectralRadius |
| (concreteMap_continuous_for_homotopy L hL) hb hw he |
|
|
| theorem exists_counterexample_of_concrete_obligations (L : ℝ) (hL : 2 < L) |
| (hC1 : IsC1SelfMap 3 (realizedMap L hL)) |
| (hentropy : Dynamics.coverEntropy (realizedMap L hL) Set.univ = (0 : EReal)) |
| (hsphere : ∃ v : RealSingularHomology 2 Sphere, v ≠ 0 ∧ |
| realHomologyMap 2 rawSphereSquare continuous_rawSphereSquare v = (2 : ℝ) • v) |
| (hbounded : BddAbove |
| (homologyEigenvalueRadii (@rawNormalForm 3) continuous_rawNormalForm)) : |
| ∃ q : ℕ, 0 < q ∧ ∃ f : CounterexampleManifold q → CounterexampleManifold q, |
| ∃ hf : IsC1SelfMap q f, |
| Dynamics.coverEntropy f (Set.univ : Set (CounterexampleManifold q)) = (0 : EReal) ∧ |
| (∃ v : RealSingularHomology 2 (CounterexampleManifold q), |
| v ≠ 0 ∧ realHomologyMap 2 f hf.1 v = (2 : ℝ) • v) ∧ |
| 0 < Real.log (2 : ℝ) ∧ |
| Real.log (2 : ℝ) ≤ Real.log (totalRealHomologySpectralRadius f hf.1) ∧ |
| ¬ Function.Bijective f ∧ 5 ≤ 2 * q + 3 := by |
| obtain ⟨v, hv, he⟩ := hsphere |
| refine ⟨3, by norm_num, realizedMap L hL, hC1, hentropy, |
| realizedMap_eigenvector_of_sphere 2 L hL 2 v hv he, |
| Real.log_pos (by norm_num), ?_, realizedMap_not_bijective L hL, by norm_num⟩ |
| exact log_two_le_realizedMap_radius_of_sphere L hL v hv he hbounded |
|
|
| end Problem340.Construction |
|
|
| end |
|
|
| end OAI |
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|
