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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 | import OAI.NumberTheory.DukePrimeDegree.Dynamics.KernelStabilizerMeasurable
import OAI.NumberTheory.DukePrimeDegree.Geometry.MeasureConjugation
namespace OAI
/-!
# The measurable root pattern of actual ergodic components
A split diagonal rescales each real root group by a nonzero scalar.
Thus it preserves the predicate that the whole root group fixes a
measure, even when it moves that measure. For the normalizer-covariant
component kernel this gives an invariant measurable event on the
component base; it does not assert that individual components are fixed.
-/
noncomputable section
open MeasureTheory ProbabilityTheory Set Filter
open scoped Topology
namespace DukePrimeDegree
open PrimeDegreePackets
lemma rootInvarianceRelation_iff_full_root {n : ℕ} {i j : Fin n}
(hij : i ≠ j) (μ : Measure (X n)) :
rootInvarianceRelation μ i j ↔ ∀ t : ℝ,
Matrix.SpecialLinearGroup.transvection hij t ∈ invariantRightSubgroup μ := by
constructor
· rintro (he | ⟨_, hroot⟩)
· exact (hij he).elim
· exact hroot
· exact fun hroot => Or.inr ⟨hij, hroot⟩
lemma all_root_mem_map_diagonal_iff {n : ℕ} {i j : Fin n}
(hij : i ≠ j) (μ : Measure (X n))
(ell : Fin n → ℝ) (hell : ∑ k, ell k = 0) (t : ℝ) :
(∀ s : ℝ, Matrix.SpecialLinearGroup.transvection hij s ∈
invariantRightSubgroup (μ.map (fun x => rightAction x
(diagonalOneParameter ell hell t)))) ↔
∀ s : ℝ, Matrix.SpecialLinearGroup.transvection hij s ∈ invariantRightSubgroup μ := by
let a := diagonalOneParameter ell hell t
let c := Real.exp ((ell i - ell j) * t)
have hc : c ≠ 0 := Real.exp_ne_zero _
have hconj (s : ℝ) : a * Matrix.SpecialLinearGroup.transvection hij s * a⁻¹ =
Matrix.SpecialLinearGroup.transvection hij (c * s) := by
rw [← diagonalOneParameter_neg]
exact diagonal_conjugation_transvection ell hell hij s t
constructor
· intro h s
have hs := (mem_invariantRightSubgroup_map_iff μ a _).mp (h (s / c))
rw [hconj, mul_div_cancel₀ _ hc] at hs
exact hs
· intro h s
apply (mem_invariantRightSubgroup_map_iff μ a _).mpr
rw [hconj]
exact h _
lemma rootInvarianceRelation_map_diagonal_iff {n : ℕ}
(μ : Measure (X n)) (ell : Fin n → ℝ) (hell : ∑ k, ell k = 0)
(t : ℝ) (i j : Fin n) :
rootInvarianceRelation (μ.map (fun x => rightAction x
(diagonalOneParameter ell hell t))) i j ↔ rootInvarianceRelation μ i j := by
by_cases hij : i = j
· simp only [rootInvarianceRelation, hij, true_or]
· rw [rootInvarianceRelation_iff_full_root hij,
rootInvarianceRelation_iff_full_root hij]
exact all_root_mem_map_diagonal_iff hij μ ell hell t
lemma measurableSet_kernel_rootInvarianceRelation
{Z : Type*} [MeasurableSpace Z] {n : ℕ}
(κ : Kernel Z (X n)) [IsFiniteKernel κ] (i j : Fin n) :
MeasurableSet {z | rootInvarianceRelation (κ z) i j} := by
by_cases hij : i = j
· simp only [rootInvarianceRelation, hij, true_or, ofPred_true]
exact MeasurableSet.univ
· have he : {z | rootInvarianceRelation (κ z) i j} =
{z | ∀ t : ℝ, Matrix.SpecialLinearGroup.transvection hij t ∈
invariantRightSubgroup (κ z)} := by
ext z
exact rootInvarianceRelation_iff_full_root hij (κ z)
rw [he]
exact measurableSet_kernel_root_invariant hij κ
/-- The actual component root pattern is unchanged along the specified
diagonal's action on the component base, almost everywhere. -/
theorem ae_component_root_pattern_diagonal {n : ℕ}
(H : Subgroup (G n)) (ell : Fin n → ℝ) (hell : ∑ k, ell k = 0) (t : ℝ)
(ha : diagonalOneParameter ell hell t ∈ Subgroup.normalizer (H : Set (G n)))
(μ : Measure (X n)) [IsProbabilityMeasure μ]
(hμ : μ.map (fun x => rightAction x (diagonalOneParameter ell hell t)) = μ)
(i j : Fin n) :
∀ᵐ z ∂μ.trim (actionInvariantSigma_le H (X n)),
rootInvarianceRelation
(condExpKernel μ (actionInvariantSigma H (X n))
(rightAction z (diagonalOneParameter ell hell t))) i j ↔
rootInvarianceRelation (condExpKernel μ (actionInvariantSigma H (X n)) z) i j := by
filter_upwards [lattice_normalizer_component_covariance H
(diagonalOneParameter ell hell t) ha μ hμ] with z hz
rw [← hz]
exact rootInvarianceRelation_map_diagonal_iff _ ell hell t i j
end DukePrimeDegree
end
end OAI
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