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720fa25 | 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 | import OAI.Dynamics.TriangleBilliards.MeasurableFlow
namespace OAI
open MeasureTheory Set
open scoped ENNReal symmDiff
noncomputable section
namespace TriangularBilliards
local instance anglePeriodPositive : Fact (0 < 2 * Real.pi) := ⟨by positivity⟩
/-- Angular Haar probability in additive angle coordinates, used to
verify invariance of the explicit angular measure. -/
def angleMeasure : Measure (AddCircle (2 * Real.pi)) :=
(ENNReal.ofReal (2 * Real.pi))⁻¹ • volume
lemma measurePreserving_toCircle :
MeasurePreserving (AddCircle.homeomorphCircle' : AddCircle (2 * Real.pi) → Circle) angleMeasure angularMeasure := by
have hc : Measurable (fun t : ℝ => (t : AddCircle (2 * Real.pi))) :=
(AddCircle.measurePreserving_mk (2 * Real.pi) 0).measurable
have he : Measurable (AddCircle.homeomorphCircle' : AddCircle (2 * Real.pi) → Circle) :=
AddCircle.homeomorphCircle'.continuous.measurable
refine ⟨he, ?_⟩
have hvol := ((AddCircle.measurePreserving_mk (2 * Real.pi) 0).smul_measure
((ENNReal.ofReal (2 * Real.pi))⁻¹)).map_eq
simp only [zero_add] at hvol
rw [angleMeasure, ← hvol, Measure.map_map he hc]
rw [angularMeasure, Measure.restrict_congr_set Ico_ae_eq_Ioc]
rfl
lemma measurePreserving_circle_mul (a : Circle) :
MeasurePreserving (fun v : Circle => a * v) angularMeasure angularMeasure := by
let θ : AddCircle (2 * Real.pi) := AddCircle.homeomorphCircle'.symm a
have hθ : AddCircle.homeomorphCircle' θ = a := AddCircle.homeomorphCircle'.apply_symm_apply a
have ha := (measurePreserving_add_left (volume : Measure (AddCircle (2 * Real.pi))) θ).smul_measure
((ENNReal.ofReal (2 * Real.pi))⁻¹)
change MeasurePreserving (fun x : AddCircle (2 * Real.pi) => θ + x) angleMeasure angleMeasure at ha
have hh := measurePreserving_toCircle.comp ha
have hf : AddCircle.homeomorphCircle' ∘ (fun x : AddCircle (2 * Real.pi) => θ + x) =
(fun v : Circle => a * v) ∘ AddCircle.homeomorphCircle' := by
funext x
exact (Real.Angle.toCircle_add θ x).trans
(congrArg (fun v : Circle => v * Real.Angle.toCircle x) hθ)
rw [hf] at hh
have hr := hh.map_of_comp (by fun_prop) measurePreserving_toCircle.measurable
rwa [measurePreserving_toCircle.map_eq] at hr
lemma measurePreserving_circle_inv :
MeasurePreserving (fun v : Circle => v⁻¹) angularMeasure angularMeasure := by
have hn := (Measure.measurePreserving_neg (volume : Measure (AddCircle (2 * Real.pi)))).smul_measure
((ENNReal.ofReal (2 * Real.pi))⁻¹)
change MeasurePreserving (fun x : AddCircle (2 * Real.pi) => -x) angleMeasure angleMeasure at hn
have hh := measurePreserving_toCircle.comp hn
have hf : AddCircle.homeomorphCircle' ∘ (fun x : AddCircle (2 * Real.pi) => -x) =
(fun v : Circle => v⁻¹) ∘ AddCircle.homeomorphCircle' := by
funext x
exact Real.Angle.toCircle_neg x
rw [hf] at hh
have hr := hh.map_of_comp (by fun_prop) measurePreserving_toCircle.measurable
rwa [measurePreserving_toCircle.map_eq] at hr
lemma reflectedDirection_eq (Q : Triangle) (i : Fin 3) (v : Circle) :
reflectedDirection Q i v = reflectedDirection Q i 1 * v⁻¹ := by
apply Subtype.ext
simp only [Circle.coe_mul, Circle.coe_inv_eq_conj]
simp [reflectedDirection, reflect]
lemma measurePreserving_reflectedDirection (Q : Triangle) (i : Fin 3) :
MeasurePreserving (reflectedDirection Q i) angularMeasure angularMeasure := by
convert (measurePreserving_circle_mul (reflectedDirection Q i 1)).comp
measurePreserving_circle_inv using 1
funext v
exact reflectedDirection_eq Q i v
lemma measurePreserving_wallReflection (Q : Triangle) (i : Fin 3) :
MeasurePreserving (wallReflection Q i) volume volume := by
have hs := measurePreserving_add_right (volume : Measure ℂ) (-Q.vertex i)
have hr := (reflectIsometry (Q.tangent i) (Q.tangent_ne_zero i)).measurePreserving
have ha := measurePreserving_add_left (volume : Measure ℂ) (Q.vertex i)
convert ha.comp (hr.comp hs) using 1
funext x
simp only [Function.comp_apply, wallReflection, sub_eq_add_neg]
rfl
/-- Ambient Liouville measure, before restriction and normalization. -/
def ambientMeasure : Measure Phase := volume.prod angularMeasure
/-- Reflection at a whole supporting line, on the ambient phase space. -/
def wallPhase (Q : Triangle) (i : Fin 3) (z : Phase) : Phase :=
(wallReflection Q i z.1, reflectedDirection Q i z.2)
lemma measurePreserving_wallPhase (Q : Triangle) (i : Fin 3) :
MeasurePreserving (wallPhase Q i) ambientMeasure ambientMeasure :=
(measurePreserving_wallReflection Q i).prod (measurePreserving_reflectedDirection Q i)
/-- The free-flight shear on the whole plane, without any boundary. -/
def freeFlow (t : ℝ) (z : Phase) : Phase := (z.1 + t • (z.2 : ℂ), z.2)
lemma measurePreserving_freeFlow (t : ℝ) :
MeasurePreserving (freeFlow t) ambientMeasure ambientMeasure := by
have hs : MeasurePreserving (fun z : Circle × ℂ => (z.1, z.2 + t • (z.1 : ℂ)))
(angularMeasure.prod volume) (angularMeasure.prod volume) := by
apply (MeasurePreserving.id angularMeasure).skew_product
(g := fun (v : Circle) (x : ℂ) => x + t • (v : ℂ))
· fun_prop
· exact Filter.Eventually.of_forall fun v =>
(measurePreserving_add_right (volume : Measure ℂ) (t • (v : ℂ))).map_eq
exact Measure.measurePreserving_swap.comp (hs.comp Measure.measurePreserving_swap)
end TriangularBilliards
end
end OAI
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