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a44267b | 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 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 | import OAI.NumberTheory.DukePrimeDegree.Geometry.ComponentRootPattern
namespace OAI
/-!
# No new invariant roots on the component base
Diagonal covariance makes each full-root invariance event invariant on
the parameter space. Diagonal ergodicity makes that event null or conull.
In the conull case, integration of the actual conditional measures gives
root invariance of the original measure. This conclusion does not assume
that the individual conditional measures are diagonal invariant.
-/
noncomputable section
open MeasureTheory ProbabilityTheory Set Filter
open scoped Topology
namespace DukePrimeDegree
open PrimeDegreePackets
lemma translation_mem_of_ae_kernel_invariant
{Z : Type*} [MeasurableSpace Z] {n : ℕ}
(τ : Measure Z) (κ : Kernel Z (X n)) (g : G n)
(hκ : ∀ᵐ z ∂τ, g ∈ invariantRightSubgroup (κ z)) :
g ∈ invariantRightSubgroup (κ ∘ₘ τ) := by
change (κ ∘ₘ τ).map (fun x => rightAction x g) = κ ∘ₘ τ
rw [Measure.map_comp τ κ (continuous_rightAction g).measurable]
apply Measure.comp_congr
filter_upwards [hκ] with z hz
change (κ z).map (fun x => rightAction x g) = κ z at hz
simpa only [Kernel.map_apply _ (continuous_rightAction g).measurable] using hz
lemma root_mem_of_ae_kernel_invariant
{Z : Type*} [MeasurableSpace Z] {n : ℕ} {i j : Fin n}
(hij : i ≠ j) (τ : Measure Z) (κ : Kernel Z (X n))
(hκ : ∀ᵐ z ∂τ, ∀ t : ℝ,
Matrix.SpecialLinearGroup.transvection hij t ∈ invariantRightSubgroup (κ z)) :
∀ t : ℝ, Matrix.SpecialLinearGroup.transvection hij t ∈
invariantRightSubgroup (κ ∘ₘ τ) := by
intro t
exact translation_mem_of_ae_kernel_invariant τ κ _ (hκ.mono fun z hz => hz t)
/-- On the actual invariant-sigma conditional measures, a root can be
present on a positive set of parameters only if it fixes the ambient
diagonal-ergodic measure. The normalizer condition concerns the subgroup
used for conditioning, not the stabilizer of each component. -/
theorem ae_component_root_implies_ambient_root
(d : ℕ) (H : Subgroup (G (d + 1)))
(hAH : ∀ u : LogCoordinates d,
diagonalFlow u ∈ Subgroup.normalizer (H : Set (G (d + 1))))
(μ : Measure (X (d + 1))) [IsProbabilityMeasure μ]
[ErgodicSMul (Multiplicative (LogCoordinates d)) (X (d + 1)) μ]
{i j : Fin (d + 1)} (hij : i ≠ j) :
∀ᵐ z ∂μ.trim (actionInvariantSigma_le H (X (d + 1))),
(∀ t : ℝ, Matrix.SpecialLinearGroup.transvection hij t ∈ invariantRightSubgroup
(condExpKernel μ (actionInvariantSigma H (X (d + 1))) z)) →
∀ t : ℝ, Matrix.SpecialLinearGroup.transvection hij t ∈ invariantRightSubgroup μ := by
let m := actionInvariantSigma H (X (d + 1))
let : MeasurableSpace (X (d + 1)) := xMeasurableSpace (d + 1)
have hm : m ≤ xMeasurableSpace (d + 1) := actionInvariantSigma_le H (X (d + 1))
let κ := condExpKernel μ m
let E : Set (X (d + 1)) := {z | ∀ t : ℝ,
Matrix.SpecialLinearGroup.transvection hij t ∈ invariantRightSubgroup (κ z)}
have hE : @MeasurableSet (X (d + 1)) m E :=
@measurableSet_kernel_root_invariant (X (d + 1)) m (d + 1) i j hij κ inferInstance
have hEinv (u : Multiplicative (LogCoordinates d)) :
(u • ·) ⁻¹' E =ᵐ[μ] E := by
let ell : Fin (d + 1) → ℝ := Fin.snoc u.toAdd (-∑ k, u.toAdd k)
have hell : ∑ k, ell k = 0 := by simp [ell, Fin.sum_univ_castSucc]
have ha : diagonalOneParameter ell hell 1 = diagonalFlow u.toAdd := by
rw [diagonalOneParameter_eq_diagonalFlow]
congr 1
funext k
simp [ell]
have hμ : μ.map (fun x => rightAction x (diagonalFlow u.toAdd)) = μ :=
(measurePreserving_smul u μ).map_eq
have hc := lattice_normalizer_component_covariance H
(diagonalFlow u.toAdd) (hAH u.toAdd) μ hμ
apply Filter.Eventually.set_eq
filter_upwards [ae_of_ae_trim hm hc] with z hz
change (∀ t : ℝ, Matrix.SpecialLinearGroup.transvection hij t ∈
invariantRightSubgroup (κ (rightAction z (diagonalFlow u.toAdd)))) ↔
∀ t : ℝ, Matrix.SpecialLinearGroup.transvection hij t ∈ invariantRightSubgroup (κ z)
change (κ z).map (fun x => rightAction x (diagonalFlow u.toAdd)) =
κ (rightAction z (diagonalFlow u.toAdd)) at hz
rw [← hz, ← ha]
exact all_root_mem_map_diagonal_iff hij (κ z) ell hell 1
have hconst := ErgodicSMul.aeconst_of_forall_preimage_smul_ae_eq
(G := Multiplicative (LogCoordinates d)) (hm E hE) hEinv
rcases eventuallyEmptyOrUniv_iff.mp hconst with hall | hnone
· have halltrim : ∀ᵐ z ∂μ.trim hm, z ∈ E := by
rw [ae_iff] at hall ⊢
change μ Eᶜ = 0 at hall
change μ.trim hm Eᶜ = 0
rwa [trim_measurableSet_eq hm hE.compl]
have hroot := @root_mem_of_ae_kernel_invariant (X (d + 1)) m (d + 1) i j
hij (μ.trim hm) κ halltrim
have hdec : κ ∘ₘ μ.trim hm = μ := condExpKernel_comp_trim hm
rw [hdec] at hroot
exact Eventually.of_forall fun _ _ => hroot
· have hnonetrim : ∀ᵐ z ∂μ.trim hm, z ∉ E := by
rw [ae_iff] at hnone ⊢
have he : {a | ¬a ∉ E} = E := by ext a; simp only [mem_ofPred_eq, not_not]
rw [he] at hnone ⊢
rwa [trim_measurableSet_eq hm hE]
exact hnonetrim.mono fun _ hz h => (hz h).elim
/-- The subgroup generated by the ambient invariant roots has the
normalizer property needed above by the explicit diagonal conjugation
formula. Thus conditioning on it introduces no new roots almost surely. -/
theorem ae_rootGenerated_component_no_new_roots
(d : ℕ) (μ : Measure (X (d + 1))) [IsProbabilityMeasure μ]
[ErgodicSMul (Multiplicative (LogCoordinates d)) (X (d + 1)) μ]
{i j : Fin (d + 1)} (hij : i ≠ j) :
∀ᵐ z ∂μ.trim (actionInvariantSigma_le (invariantRootGeneratedGroup μ) (X (d + 1))),
(∀ t : ℝ, Matrix.SpecialLinearGroup.transvection hij t ∈ invariantRightSubgroup
(condExpKernel μ (actionInvariantSigma (invariantRootGeneratedGroup μ) (X (d + 1))) z)) →
∀ t : ℝ, Matrix.SpecialLinearGroup.transvection hij t ∈ invariantRightSubgroup μ := by
apply ae_component_root_implies_ambient_root d (invariantRootGeneratedGroup μ) ?_ μ hij
intro u
let ell : Fin (d + 1) → ℝ := Fin.snoc u (-∑ k, u k)
have hell : ∑ k, ell k = 0 := by simp [ell, Fin.sum_univ_castSucc]
have ha : diagonalOneParameter ell hell 1 = diagonalFlow u := by
rw [diagonalOneParameter_eq_diagonalFlow]
congr 1
funext k
simp [ell]
rw [← ha]
exact diagonal_normalizes_rootGeneratedGroup μ ell hell 1
/-- Roots in the conditioning subgroup also preserve the conditional
measures. Countably many rational times suffice because each actual
measure stabilizer is closed. -/
theorem ae_conditional_root_invariant_of_subgroup_root
{n : ℕ} (H : Subgroup (G n)) (μ : Measure (X n)) [IsProbabilityMeasure μ]
(hH : H ≤ invariantRightSubgroup μ) {i j : Fin n} (hij : i ≠ j)
(hU : ∀ t : ℝ, Matrix.SpecialLinearGroup.transvection hij t ∈ H) :
∀ᵐ z ∂μ.trim (actionInvariantSigma_le H (X n)),
∀ t : ℝ, Matrix.SpecialLinearGroup.transvection hij t ∈ invariantRightSubgroup
(condExpKernel μ (actionInvariantSigma H (X n)) z) := by
have hq (q : ℚ) : ∀ᵐ z ∂μ.trim (actionInvariantSigma_le H (X n)),
Matrix.SpecialLinearGroup.transvection hij (q : ℝ) ∈ invariantRightSubgroup
(condExpKernel μ (actionInvariantSigma H (X n)) z) := by
have hfix : actionInvariantSigma H (X n) ≤ MeasurableSpace.invariants
(fun x => rightAction x (Matrix.SpecialLinearGroup.transvection hij (q : ℝ))) := by
have hh := iInf_le (fun h : H => MeasurableSpace.invariants (fun x : X n => h • x))
(⟨(Matrix.SpecialLinearGroup.transvection hij (q : ℝ))⁻¹, H.inv_mem (hU q)⟩ : H)
change actionInvariantSigma H (X n) ≤ MeasurableSpace.invariants
(fun x => rightAction x ((Matrix.SpecialLinearGroup.transvection hij (q : ℝ))⁻¹)⁻¹) at hh
simpa only [inv_inv] using hh
exact condExpKernel_invariant (actionInvariantSigma H (X n)) μ
(actionInvariantSigma_le H (X n))
⟨(continuous_rightAction _).measurable, hH (hU q)⟩ hfix
filter_upwards [ae_all_iff.mpr hq] with z hz
exact (all_root_mem_iff_rational_root_mem hij _).mpr hz
/-- The full-root pattern of the actual invariant-sigma conditional
measures agrees almost everywhere with the ambient pattern. This is
stronger than a measurable root-pattern assertion and still does not
assert diagonal invariance or homogeneity of individual components. -/
theorem ae_rootGenerated_component_root_iff
(d : ℕ) (μ : Measure (X (d + 1))) [IsProbabilityMeasure μ]
[ErgodicSMul (Multiplicative (LogCoordinates d)) (X (d + 1)) μ]
{i j : Fin (d + 1)} (hij : i ≠ j) :
∀ᵐ z ∂μ.trim (actionInvariantSigma_le (invariantRootGeneratedGroup μ) (X (d + 1))),
(∀ t : ℝ, Matrix.SpecialLinearGroup.transvection hij t ∈ invariantRightSubgroup
(condExpKernel μ (actionInvariantSigma (invariantRootGeneratedGroup μ) (X (d + 1))) z)) ↔
∀ t : ℝ, Matrix.SpecialLinearGroup.transvection hij t ∈ invariantRightSubgroup μ := by
by_cases hroot : ∀ t : ℝ,
Matrix.SpecialLinearGroup.transvection hij t ∈ invariantRightSubgroup μ
· have hc := ae_conditional_root_invariant_of_subgroup_root (invariantRootGeneratedGroup μ)
μ (invariantRootGeneratedGroup_le μ) hij
(root_mem_invariantRootGeneratedGroup μ hij (Or.inr ⟨hij, hroot⟩))
exact hc.mono fun _ hz => ⟨fun _ => hroot, fun _ => hz⟩
· exact (ae_rootGenerated_component_no_new_roots d μ hij).mono
fun _ hz => ⟨hz, fun h => (hroot h).elim⟩
end DukePrimeDegree
end
end OAI
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