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695bd70 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 | import OAI.Geometry.SingularHomology.MapHomology
import OAI.Dynamics.EntropyCounterexample.RealizedConstruction
import OAI.Dynamics.EntropyCounterexample.HomotopySpectrum
import OAI.Dynamics.EntropyCounterexample.SpectralRadius
namespace OAI
noncomputable section
namespace Problem340.Construction
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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