File size: 3,052 Bytes
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 | import OAI.NumberTheory.DukePrimeDegree.Entropy.RootInformationBound
import OAI.NumberTheory.DukePrimeDegree.Dynamics.RootInvariantLeaves
namespace OAI
/-! # Mean root coefficients and their elementary properties
The coefficient is the mean leaf information at unit scale. Diagonal
covariance makes real-time mean information linear in time.
-/
noncomputable section
open MeasureTheory Filter Set
open scoped Topology ENNReal
namespace DukePrimeDegree
open PrimeDegreePackets
/-- The actual mean information at unit scale, with zero diagonal. -/
def canonicalRootCoefficient (d : ℕ) (μ : Measure (X (d + 1))) [IsFiniteMeasure μ]
(i j : Fin (d + 1)) : ℝ :=
if hij : i = j then 0 else (rootInformationMean hij μ 1).toReal
variable (d : ℕ) (μ : Measure (X (d + 1))) [IsProbabilityMeasure μ]
(hA : ∀ u : LogCoordinates d, μ.map (fun x => rightAction x (diagonalFlow u)) = μ)
lemma canonicalRootCoefficient_diagonal (i : Fin (d + 1)) :
canonicalRootCoefficient d μ i i = 0 := by simp [canonicalRootCoefficient]
lemma canonicalRootCoefficient_nonneg (i j : Fin (d + 1)) :
0 ≤ canonicalRootCoefficient d μ i j := by
unfold canonicalRootCoefficient
split_ifs
· exact le_rfl
· exact ENNReal.toReal_nonneg
include hA
lemma canonicalRootCoefficient_le_one (i j : Fin (d + 1)) :
canonicalRootCoefficient d μ i j ≤ 1 := by
unfold canonicalRootCoefficient
split_ifs with hij
· exact zero_le_one
· exact (ENNReal.toReal_mono ENNReal.one_ne_top
(rootInformationMean_le_one d hij μ hA)).trans_eq ENNReal.toReal_one
lemma canonicalRootCoefficient_zero_iff (i j : Fin (d + 1)) (hij : i ≠ j) :
canonicalRootCoefficient d μ i j = 0 ↔
∀ᵐ x ∂μ, intrinsicRootLeaf hij μ x = Measure.dirac 0 := by
rw [canonicalRootCoefficient, dite_eq_right hij, ENNReal.toReal_eq_zero_iff]
have hfin : rootInformationMean hij μ 1 ≠ ∞ :=
ne_of_lt ((rootInformationMean_le_one d hij μ hA).trans_lt ENNReal.one_lt_top)
simp only [hfin, or_false]
exact rootInformationMean_eq_zero_iff_dirac d hij μ hA
lemma canonicalRootCoefficient_one_of_invariant (i j : Fin (d + 1)) (hij : i ≠ j)
(hinv : ∀ t : ℝ, μ.map (fun x => rightAction x
(Matrix.SpecialLinearGroup.transvection hij t)) = μ) :
canonicalRootCoefficient d μ i j = 1 := by
rw [canonicalRootCoefficient, dite_eq_right hij, rootInformationMean_of_invariant d hij μ hA hinv]
norm_num
/-- Actual real-time mean information is the root coefficient times time. -/
lemma rootInformationMean_eq_coefficient (i j : Fin (d + 1)) (hij : i ≠ j)
{t : ℝ} (ht : 0 ≤ t) :
rootInformationMean hij μ t =
ENNReal.ofReal (t * canonicalRootCoefficient d μ i j) := by
have hfin : rootInformationMean hij μ 1 ≠ ∞ :=
ne_of_lt ((rootInformationMean_le_one d hij μ hA).trans_lt ENNReal.one_lt_top)
rw [rootInformationMean_real d hij μ hA ht, canonicalRootCoefficient, dite_eq_right hij,
ENNReal.ofReal_mul ht, ENNReal.ofReal_toReal hfin]
end DukePrimeDegree
end
end OAI
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