File size: 2,891 Bytes
f1f7953 | 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 | import OAI.Analysis.IrrationalRotation.Model.MixedRadixCircle
import OAI.Analysis.IrrationalRotation.Sampling.MixedRadix
import OAI.Analysis.IrrationalRotation.Analysis.DensityTransport
namespace OAI
open MeasureTheory
open scoped ENNReal
namespace Invariant233.MixedRadix
theorem measurePreserving_unitCircle :
MeasurePreserving (fun x : ℝ => (x : AddCircle (1 : ℝ)))
(volume.restrict (Set.Ico 0 1)) volume := by
rw [restrict_Ico_eq_restrict_Ioc]
simpa using AddCircle.measurePreserving_mk (1 : ℝ) 0
theorem measurable_circleValue {r : ℕ → ℕ} (hr : ∀ n, 2 ≤ r n) :
Measurable (@circleValue r) :=
AddCircle.measurable_mk'.comp (measurable_ofDigits hr)
theorem measurable_circleCanonical (r : ℕ → ℕ) (hr : ∀ n, 0 < r n) :
Measurable (circleCanonical r hr) :=
(measurable_canonicalDigits r hr).comp
(measurable_subtype_coe.comp (AddCircle.measurableEquivIco (1 : ℝ) 0).measurable)
theorem circleValue_circleCanonical {r : ℕ → ℕ} (hr : ∀ n, 2 ≤ r n)
(hp : ∀ n, 0 < r n) (x : AddCircle (1 : ℝ)) :
circleValue (circleCanonical r hp x) = x := by
have hx : (AddCircle.equivIco (1 : ℝ) 0 x : ℝ) ∈ Set.Ico 0 1 := by
simpa using (AddCircle.equivIco (1 : ℝ) 0 x).property
unfold circleValue circleCanonical
rw [ofDigits_canonicalDigits hr hx]
exact AddCircle.coe_equivIco
theorem circleCanonical_circleValue {r : ℕ → ℕ} (hp : ∀ n, 0 < r n)
(d : Digits r) (hnt : Nonterminal d) :
circleCanonical r hp (circleValue d) = d := by
have hx : ofDigits d ∈ Set.Ico (0 : ℝ) (0 + 1) := by
simpa using ofDigits_mem_Ico hp d hnt
unfold circleCanonical circleValue
rw [AddCircle.equivIco_coe_eq hx]
exact canonicalDigits_ofDigits hp d hnt
theorem circleValue_measurePreserving {r : ℕ → ℕ} (hr : ∀ n, 2 ≤ r n)
(μ : Measure (Digits r))
(hlaw : μ.map ofDigits = volume.restrict (Set.Ico 0 1)) :
MeasurePreserving (@circleValue r) μ volume := by
exact measurePreserving_unitCircle.comp ⟨measurable_ofDigits hr, hlaw⟩
theorem circle_density_transport {r : ℕ → ℕ} (hr : ∀ n, 2 ≤ r n)
(hp : ∀ n, 0 < r n) (μ : Measure (Digits r))
(hlaw : μ.map ofDigits = volume.restrict (Set.Ico 0 1))
(hnt : ∀ᵐ d ∂μ, Nonterminal d) {h : Digits r → ℝ} (hh : Measurable h) :
(μ.withDensity (fun d => ENNReal.ofReal (h d))).map circleValue =
volume.withDensity (fun x : AddCircle (1 : ℝ) =>
ENNReal.ofReal (h (circleCanonical r hp x))) ∧
eLpNorm (h ∘ circleCanonical r hp) 2 volume = eLpNorm h 2 μ := by
apply DensityTransport.real_density_transport (measurable_circleValue hr)
(measurable_circleCanonical r hp) (circleValue_measurePreserving hr μ hlaw).map_eq
· filter_upwards [hnt] with d hd
exact circleCanonical_circleValue hp d hd
· exact hh
end Invariant233.MixedRadix
end OAI
|