import OAI.Dynamics.TriangleBilliards.SpatialConstancy namespace OAI universe uAlpha uBeta open MeasureTheory Set open scoped ENNReal symmDiff noncomputable section open MeasureTheory Set Filter Function Metric open scoped Topology Convolution ContDiff noncomputable section open MeasureTheory Set open scoped ENNReal noncomputable section open MeasureTheory Set Filter BoundedContinuousFunction open scoped ENNReal Topology ComplexConjugate noncomputable section open MeasureTheory Set Filter open scoped Topology ComplexConjugate noncomputable section open MeasureTheory Filter open scoped ComplexConjugate noncomputable section open MeasureTheory Filter Set open scoped Topology ComplexConjugate noncomputable section open Filter Finset Set open scoped Topology BigOperators noncomputable section open MeasureTheory Filter Set open scoped Topology ContDiff NNReal open MeasureTheory Filter Set open scoped Topology ComplexConjugate noncomputable section open Filter Set open scoped Topology noncomputable section open MeasureTheory Set Filter open scoped Topology noncomputable section namespace TriangularBilliards open Analysis Analytic SpatialSmoothing Filter Set open scoped Topology ContDiff ComplexConjugate def spatialMean (Q : Triangle) (f : DoublePhase → ℂ) (c : DirectionParity) : ℂ := ∫ x, f ((x,c.1),c.2) ∂((volume Q.table)⁻¹ • volume.restrict Q.table) lemma spatialMean_stronglyMeasurable (Q : Triangle) {f : DoublePhase → ℂ} (hf : StronglyMeasurable f) : StronglyMeasurable (spatialMean Q f) := by have h : StronglyMeasurable (Function.uncurry (fun c : DirectionParity => fun x : ℂ => f ((x,c.1),c.2))) := hf.comp_measurable ((measurable_snd.prodMk measurable_fst.fst).prodMk measurable_fst.snd) exact h.integral_prod_right lemma Triangle.clearance_pos_of_mem_side (Q : Triangle) {q : ℂ} {i : Fin 3} (hq : q ∈ Q.side i) : 0 < Q.clearance q := by apply ((finite_range Q.vertex).isClosed.notMem_iff_infDist_pos (range_nonempty _)).mp rintro ⟨k,rfl⟩ obtain ⟨j,hjk,hji⟩ := (by decide : ∀ i k : Fin 3, ∃ j : Fin 3, j≠k ∧ j≠i+2) i k have hj : Q.basis.coord j (Q.vertex k)=0 := (Q.basis.coord_apply j k).trans (ite_eq_right hjk) exact hji ((Q.side_coord_zero_iff hq j).mp hj) lemma Triangle.side_mem_hull (Q : Triangle) {q : ℂ} {i : Fin 3} (hq : q ∈ Q.side i) : q ∈ convexHull ℝ (range Q.vertex) := (convex_convexHull ℝ (range Q.vertex)).segment_subset (subset_convexHull ℝ _ (mem_range_self i)) (subset_convexHull ℝ _ (mem_range_self (i+1))) (openSegment_subset_segment ℝ _ _ hq) lemma directS_eq_const_of_slice (Q : Triangle) {ε : ℝ} (hε : 0 < ε) {f : DoublePhase → ℂ} {v : Circle} {b : ZMod 2} {a : ℂ} (hf : (fun y => f ((y,v),b)) =ᵐ[volume.restrict Q.table] (fun _ => a)) {x : ℂ} (hx : Metric.ball x ε ⊆ Q.table) : directS Q ε f b v x = a := by have hi : (∫ y in Q.table, kernel ε (x-y)) = 1 := by rw [← integral_indicator Q.measurableSet_table] calc _ = ∫ y : ℂ, kernel ε (x-y) := by apply integral_congr_ae filter_upwards with y by_cases hy : y∈Q.table · exact indicator_of_mem hy _ · rw [indicator_of_notMem hy] symm by_contra hn have hm : x-y ∈ support (kernel ε) := hn rw [kernel_support hε,mem_ball_zero_iff] at hm apply hy apply hx simpa only [Metric.mem_ball,dist_eq_norm,norm_sub_rev] using hm _ = 1 := (integral_sub_left_eq_self (fun y : ℂ => kernel ε y) volume x).trans (kernel_integral hε) calc _ = ∫ y in Q.table, kernel ε (x-y) • a := by apply integral_congr_ae filter_upwards [hf] with y hy rw [hy] _ = a := by rw [integral_smul_const,hi,one_smul] lemma reflectedSmoothing_eq_const_of_slice (Q : Triangle) {ε : ℝ} (hε : 0 < ε) {f : DoublePhase → ℂ} {v : Circle} {b : ZMod 2} {a : ℂ} (hf : (fun y => f ((y,v),b)) =ᵐ[volume.restrict Q.table] (fun _ => a)) {x : ℂ} (hx : Metric.ball x ε ⊆ Q.table) (hcoord : ∀ j : Fin 3, Q.coordinateBound*ε < Q.basis.coord (j+2) x) : reflectedSmoothing Q ε f ((x,v),b) = a := by rw [reflectedSmoothing,directS_eq_const_of_slice Q hε hf hx] have hz : (∑ j : Fin 3, reflectedS Q j ε f b v x)=0 := Finset.sum_eq_zero (fun j _ => reflectedS_zero_of_coord Q j hε f b v (hcoord j)) rw [hz,add_zero] end TriangularBilliards open MeasureTheory Set Filter open scoped Topology ComplexConjugate ContDiff noncomputable section local instance : Fact (Module.finrank ℝ ℂ = 2) := ⟨Complex.finrank_real_complex⟩ lemma equal_of_zero_gradient_openSegment {F : ℂ → ℂ} (hF : ContDiff ℝ ∞ F) {x y : ℂ} (hD : ∀ z ∈ openSegment ℝ x y, fderiv ℝ F z=0) : F x=F y := by have hcl := closure_minimal hD (isClosed_eq (hF.continuous_fderiv (by simp)) continuous_const) rw [closure_openSegment] at hcl have hl := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le (f := F) (f' := fun _ => (0 : ℂ →L[ℝ] ℂ)) (C := (0:ℝ)) (fun w hw => (hcl hw ▸ (hF.differentiable (by simp) w).hasFDerivAt).hasFDerivWithinAt) (fun _ _ => by simp) (convex_segment x y) (left_mem_segment ℝ x y) (right_mem_segment ℝ x y) exact (sub_eq_zero.mp (norm_le_zero_iff.mp (by simpa only [zero_mul] using hl))).symm lemma irrational_arg_of_angle {a b : ℂ} (ha : a≠0) (hb : b≠0) (h : Irrational (InnerProductGeometry.angle a b / Real.pi)) : Irrational ((conj a*b).arg / Real.pi) := by have he : InnerProductGeometry.angle a b = |(conj a*b).arg| := by rw [Complex.orientation.angle_eq_abs_oangle_toReal ha hb,Complex.oangle,Complex.arg_coe_angle_toReal_eq_arg] rw [he] at h rcases le_or_gt 0 (conj a*b).arg with hn|hn · simpa only [abs_of_nonneg hn] using h · simpa only [abs_of_neg hn,neg_div,irrational_neg_iff] using h lemma complex_div_conj_eq_exp_double_arg {w : ℂ} (hw : w≠0) : w / conj w = Complex.exp (2*(w.arg:ℂ)*Complex.I) := by have hn : (‖w‖:ℂ) ≠ 0 := Complex.ofReal_ne_zero.mpr (norm_ne_zero_iff.mpr hw) have hconj : conj w = (‖w‖:ℂ)*Complex.exp (-(w.arg:ℂ)*Complex.I) := by conv_lhs => rw [← Complex.norm_mul_exp_arg_mul_I w] simp only [map_mul,Complex.conj_ofReal,← Complex.exp_conj,Complex.conj_I] congr 2; ring rw [hconj] conv_lhs => lhs; rw [← Complex.norm_mul_exp_arg_mul_I w] rw [mul_div_mul_left _ _ hn,← Complex.exp_sub] congr 1 ring namespace TriangularBilliards open Analysis Analytic SpatialSmoothing Filter Set open scoped Topology ContDiff ComplexConjugate lemma spatialMean_reflection_invariant (Q : Triangle) {f : DoublePhase → ℂ} (hfm : StronglyMeasurable f) (hf : MemLp f 2 (doubleMeasure Q)) {H : ℝ} (hb : ∀ z, ‖f z‖ ≤ H) (hfx : (geodesicHilbertFlow Q).HasGenerator (hf.toLp f) 0) (i : Fin 3) : spatialMean Q f ∘ directionParityReflection Q i =ᵐ[angularMeasure.prod parityMeasure] spatialMean Q f := by obtain ⟨x,hx⟩ := Q.table_nonempty let q := midpoint ℝ (Q.vertex i) (Q.vertex (i+1)) have hq : q ∈ Q.side i := midpoint_mem_openSegment _ _ have hopen : openSegment ℝ x q ⊆ Q.table := (convex_convexHull ℝ (range Q.vertex)).openSegment_interior_self_subset_interior hx (Q.side_mem_hull hq) have hpos : ∀ y ∈ segment ℝ x q, 0 < Q.clearance y := by intro y hy rw [← insert_endpoints_openSegment] at hy rcases hy with rfl|rfl|hy · exact Q.clearance_pos_of_mem_table hx · exact Q.clearance_pos_of_mem_side hq · exact Q.clearance_pos_of_mem_table (hopen hy) have hscompact : IsCompact (segment ℝ x q) := by rw [segment_eq_image_lineMap] exact isCompact_Icc.image (by fun_prop) obtain ⟨y,hy,hmin⟩ := hscompact.exists_isMinOn ⟨x,left_mem_segment ℝ x q⟩ Q.clearance_lipschitz.continuous.continuousOn obtain ⟨r,hr,hrball⟩ := Metric.mem_nhds_iff.mp (Q.isOpen_table.mem_nhds hx) let K := Q.safetyFactor+1 have hK : 0 < K := by dsimp [K]; linarith [Q.safetyFactor_pos] have hev (a d : ℝ) (hd : 0 < d) : ∀ᶠ ε : ℝ in 𝓝 0, a*ε < d := by have hc : Continuous (fun ε : ℝ => a*ε) := by fun_prop simpa only [mul_zero] using hc.continuousAt.eventually (gt_mem_nhds (by simpa only [mul_zero] using hd)) have hnb : ∀ᶠ ε : ℝ in 𝓝 0, 2*(K*ε)*Q.coordinateBound < 1 ∧ (2*K+1)*ε < Q.clearance y ∧ ε hev Q.coordinateBound _ ((Q.table_iff_coords x).mp hx (j+2)))] with ε h₁ h₂ h₃ h₄ exact ⟨by nlinarith,h₂,by simpa using h₃,h₄⟩ have hnb' : ∀ᶠ ε : ℝ in 𝓝[>] 0, 0 < ε ∧ 2*(K*ε)*Q.coordinateBound < 1 ∧ (2*K+1)*ε < Q.clearance y ∧ ε] 0, 0 < ε from self_mem_nhdsWithin).and (hnb.filter_mono nhdsWithin_le_nhds) obtain ⟨ε,hε,hsmall,hsafe,her,hcoord⟩ := hnb'.exists let R := K*ε let A := (2*K+1)*ε have hR : Q.safetyFactor*ε < R := by dsimp [R,K]; nlinarith have hA : 2*R+ε ≤ A := by dsimp [R,A]; ring_nf; rfl have hseg : ∀ z∈segment ℝ x q, A < Q.clearance z := fun z hz => hsafe.trans_le (hmin hz) have hseam : Q.safetyFactor*ε < Q.clearance q := by have hh := hseg q (right_mem_segment ℝ x q) dsimp [A,K] at hh nlinarith [Q.safetyFactor_pos] have hball : Metric.ball x ε ⊆ Q.table := (Metric.ball_subset_ball her.le).trans hrball have hconst := bounded_invariant_spatial_constancy Q hfm hf hb hfx have hD := double_ae_slices Q (invariant_smoothing_gradient_zero Q hε hR hsmall hA hfm hf hfx hb) filter_upwards [hconst,hD,(measurePreserving_directionParity Q i).quasiMeasurePreserving.ae hconst, (measurePreserving_directionParity Q i).quasiMeasurePreserving.ae hD] with c hc hd hrc hrd have he (c : DirectionParity) (hc : (fun z => f ((z,c.1),c.2)) =ᵐ[(volume Q.table)⁻¹ • volume.restrict Q.table] (fun _ => spatialMean Q f c)) (hd : ∀ᵐ z ∂((volume Q.table)⁻¹ • volume.restrict Q.table), A ≤ Q.clearance z → fderiv ℝ (fun w => reflectedSmoothing Q ε f ((w,c.1),c.2)) z=0) : reflectedSmoothing Q ε f ((q,c.1),c.2) = spatialMean Q f c := by have hsm := reflectedSmoothing_contDiff Q hε (bounded_spatialSlice_integrable Q hfm hb c.1 c.2) (fun j => bounded_spatialSlice_integrable Q hfm hb (reflectedDirection Q j c.1) (c.2+1)) have hdx := smoothing_gradient_zero_on_slice Q hε hfm hb c hd have hxy := equal_of_zero_gradient_openSegment hsm (fun z hz => hdx z (hopen hz) (hseg z (openSegment_subset_segment ℝ x q hz))) rw [← hxy] exact reflectedSmoothing_eq_const_of_slice Q hε ((Measure.ae_ennreal_smul_measure_iff (by simpa using Q.area_lt_top.ne)).mp hc) hball hcoord have hs := (reflectedSmoothing_seam_germ Q hq hε hseam f c.2 c.1).self_of_nhds rw [wallReflection_fixed_side Q hq] at hs change spatialMean Q f (directionParityReflection Q i c) = spatialMean Q f c rw [← he _ hrc hrd,← he _ hc hd] exact hs end TriangularBilliards namespace TriangularBilliards open scoped ComplexConjugate lemma reflectedDirection_pair (Q : Triangle) (i : Fin 3) (v : Circle) : reflectedDirection Q (i+2) (reflectedDirection Q i v) = Circle.exp (2 * (conj (Q.vertex (i+1)-Q.vertex i) * (Q.vertex (i+2)-Q.vertex i)).arg) * v := by let a := Q.vertex (i+1)-Q.vertex i let b := Q.vertex (i+2)-Q.vertex i have ha : a ≠ 0 := Q.tangent_ne_zero i have hb : b ≠ 0 := sub_ne_zero.mpr (Q.nondegenerate.injective.ne (by omega)) have he : Q.tangent (i+2) = -b := by dsimp [Triangle.tangent,b] rw [show i+2+1=i by omega] abel have hw : conj a*b ≠ 0 := mul_ne_zero ((map_ne_zero (starRingEnd ℂ)).mpr ha) hb apply Subtype.ext change reflect (Q.tangent (i+2)) (reflect (Q.tangent i) (v:ℂ)) = _ rw [he] change reflect (-b) (reflect a (v:ℂ)) = _ simp only [reflect,Complex.star_def,map_mul,map_div₀,map_neg,RingHomCompTriple.comp_apply, RingHom.id_apply,Circle.coe_mul,Circle.coe_exp,Complex.ofReal_mul,Complex.ofReal_ofNat] rw [← complex_div_conj_eq_exp_double_arg hw] simp only [map_mul,Complex.conj_conj] field_simp end TriangularBilliards namespace TriangularBilliards open MeasureTheory Set Filter open scoped Topology ComplexConjugate local instance : Fact (0 < 2 * Real.pi) := ⟨by positivity⟩ lemma angle_irrational_ergodic {a : ℝ} (ha : Irrational (a / Real.pi)) : Ergodic (fun x : AddCircle (2*Real.pi) => (2*a : ℝ) + x) ((ENNReal.ofReal (2*Real.pi))⁻¹ • volume) := by let : IsFiniteMeasure (((ENNReal.ofReal (2*Real.pi))⁻¹) • (volume : Measure (AddCircle (2*Real.pi)))) := Measure.smul_finite volume (ENNReal.inv_ne_top.mpr (by positivity)) apply ergodic_add_left_of_denseRange_zsmul apply AddCircle.denseRange_zsmul_coe_iff.mpr convert ha using 1 field_simp lemma angular_reflections_constant (Q : Triangle) (hQ : Q.HasIrrationalAngle) {g : Circle → ℂ} (hg : StronglyMeasurable g) (hi : ∀ i : Fin 3, g ∘ reflectedDirection Q i =ᵐ[angularMeasure] g) : ∃ a : ℂ, g =ᵐ[angularMeasure] fun _ => a := by obtain ⟨i,hiirr⟩ := hQ let a := Q.vertex (i+1)-Q.vertex i let b := Q.vertex (i+2)-Q.vertex i let θ := (conj a*b).arg have ha : a≠0 := Q.tangent_ne_zero i have hb : b≠0 := sub_ne_zero.mpr (Q.nondegenerate.injective.ne (by omega)) have hθ : Irrational (θ/Real.pi) := irrational_arg_of_angle ha hb hiirr have hinv : (fun v => g (Circle.exp (2*θ)*v)) =ᵐ[angularMeasure] g := by filter_upwards [hi i,(measurePreserving_reflectedDirection Q i).quasiMeasurePreserving.ae (hi (i+2))] with v hv hrv rw [Function.comp_apply,reflectedDirection_pair] at hrv exact hrv.trans hv let e := AddCircle.homeomorphCircle' have hgm : StronglyMeasurable (g ∘ e) := hg.comp_measurable measurePreserving_toCircle.measurable have hie : (g ∘ e) ∘ (fun x : AddCircle (2*Real.pi) => (2*θ : ℝ)+x) =ᵐ[angleMeasure] g ∘ e := by filter_upwards [measurePreserving_toCircle.quasiMeasurePreserving.ae hinv] with x hx have hadd : e (((2*θ : ℝ) : AddCircle (2*Real.pi))+x) = Circle.exp (2*θ)*e x := (Real.Angle.toCircle_add (((2*θ : ℝ) : AddCircle (2*Real.pi))) x).trans (congrArg (fun z : Circle => z * e x) (Real.Angle.toCircle_coe (2*θ))) change g (e (((2*θ : ℝ) : AddCircle (2*Real.pi))+x)) = g (e x) rw [hadd] exact hx obtain ⟨c,hc⟩ := (angle_irrational_ergodic hθ).ae_eq_const_of_ae_eq_comp_ae hgm.aestronglyMeasurable hie refine ⟨c,?_⟩ rw [← measurePreserving_toCircle.map_eq] exact (ae_map_iff measurePreserving_toCircle.measurable.aemeasurable (measurableSet_eq_fun hg.measurable measurable_const)).mpr hc end TriangularBilliards lemma ae_fst_of_ae_prod {α : Type uAlpha} {β : Type uBeta} [MeasurableSpace α] [MeasurableSpace β] {μ : Measure α} {ν : Measure β} [SFinite ν] [NeZero ν] {p : α → Prop} (h : ∀ᵐ z ∂μ.prod ν, p z.1) : ∀ᵐ x ∂μ, p x := by filter_upwards [Measure.ae_ae_of_ae_prod h] with x hx obtain ⟨y,hy⟩ := hx.exists exact hy lemma measure_zero_or_one_of_indicator_constant {α : Type uAlpha} [MeasurableSpace α] {μ : Measure α} [IsProbabilityMeasure μ] {A : Set α} {c : ℂ} (h : A.indicator (fun _ => (1 : ℂ)) =ᵐ[μ] fun _ => c) : μ A=0 ∨ μ A=1 := by classical obtain ⟨x,hx⟩ := h.exists by_cases hxA : x∈A · have hc : c=1 := by simpa only [indicator_of_mem hxA] using hx.symm right have he : A =ᵐ[μ] univ := by filter_upwards [h] with y hy apply propext simp only [mem_univ,iff_true] by_contra hn rw [indicator_of_notMem hn,hc] at hy exact zero_ne_one hy rw [measure_congr he,measure_univ] · have hc : c=0 := by simpa only [indicator_of_notMem hxA] using hx.symm left apply measure_eq_zero_iff_ae_notMem.mpr filter_upwards [h] with y hy intro hym rw [indicator_of_mem hym,hc] at hy exact one_ne_zero hy lemma invariant_indicator {α : Type uAlpha} [MeasurableSpace α] (μ : Measure α) (T : α → α) (A : Set α) (h : μ ((T ⁻¹' A) ∆ A)=0) : (fun x => A.indicator (fun _ => (1:ℂ)) (T x)) =ᵐ[μ] A.indicator (fun _ => (1:ℂ)) := by classical filter_upwards [measure_symmDiff_eq_zero_iff.mp h] with x hx change (T x∈A) = (x∈A) at hx by_cases hm : x∈A · have ht : T x∈A := by rwa [hx] simp only [indicator_of_mem hm,indicator_of_mem ht] · have ht : T x∉A := by rwa [hx] simp only [indicator_of_notMem hm,indicator_of_notMem ht] namespace TriangularBilliards open Analysis Analytic SpatialSmoothing Filter Set open scoped Topology /-- Every bounded measurable invariant of the actual billiard is constant. Spatial rigidity is proved above on the oriented double; side holonomy now uses the actual irrational angle, with no Diophantine rate assumption. -/ theorem bounded_billiard_invariant_constant (Q : Triangle) (hQ : Q.HasIrrationalAngle) {f : Phase → ℂ} (hfm : StronglyMeasurable f) (hf : MemLp f 2 (phaseMeasure Q)) {H : ℝ} (hb : ∀ z, ‖f z‖ ≤ H) (hinv : ∀ t : ℝ, (fun z => f (billiardFlow Q t z)) =ᵐ[phaseMeasure Q] f) : ∃ c : ℂ, f =ᵐ[phaseMeasure Q] fun _ => c := by let F : DoublePhase → ℂ := fun z => f z.1 have hFm : StronglyMeasurable F := hfm.comp_measurable measurable_fst have mp : MeasurePreserving (Prod.fst : DoublePhase → Phase) (doubleMeasure Q) (phaseMeasure Q) := measurePreserving_fst have hF : MemLp F 2 (doubleMeasure Q) := hf.comp_measurePreserving mp have hFi : (geodesicHilbertFlow Q).HasGenerator (hF.toLp F) 0 := by apply ((geodesicHilbertFlow Q).hasGenerator_zero_iff _).mpr intro t apply Lp.ext filter_upwards [geodesic_koopman_ae Q t (hF.toLp F),hF.coeFn_toLp, (measurePreserving_doubleFlow Q t).quasiMeasurePreserving.ae hF.coeFn_toLp, mp.quasiMeasurePreserving.ae (hinv t)] with z hz hfz hftz hiz rw [hz,hftz,hfz] exact hiz let g : Circle → ℂ := fun v => spatialMean Q F (v,0) have hgm : StronglyMeasurable g := (spatialMean_stronglyMeasurable Q hFm).comp_measurable (measurable_id.prodMk measurable_const) have hgmean (c : DirectionParity) : spatialMean Q F c = g c.1 := rfl have hgi (i : Fin 3) : g ∘ reflectedDirection Q i =ᵐ[angularMeasure] g := by apply ae_fst_of_ae_prod (ν := parityMeasure) have hh := spatialMean_reflection_invariant Q hFm hF (fun z => hb z.1) hFi i filter_upwards [hh] with z hz exact hz obtain ⟨c,hgc⟩ := angular_reflections_constant Q hQ hgm hgi have hs := bounded_invariant_spatial_constancy Q hFm hF (fun z => hb z.1) hFi have hs' : ∀ᵐ v ∂angularMeasure, (fun x => f (x,v)) =ᵐ[(volume Q.table)⁻¹ • volume.restrict Q.table] fun _ => g v := by apply ae_fst_of_ae_prod (ν := parityMeasure) exact hs have hh : ∀ᵐ v ∂angularMeasure, ∀ᵐ x ∂((volume Q.table)⁻¹ • volume.restrict Q.table), f (x,v)=c := by filter_upwards [hs',hgc] with v hv hvc filter_upwards [hv] with x hx exact hx.trans hvc refine ⟨c,?_⟩ have hm : MeasurableSet {z : ℂ × Circle | f z=c} := measurableSet_eq_fun hfm.measurable measurable_const apply (Measure.ae_prod_iff_ae_ae hm).mpr exact (Measure.ae_ae_comm (p := fun x v => f (x,v)=c) hm).mpr hh /-- The invariant-set endpoint, for invariance modulo null sets at every real (time), on the normalized interior phase space of every irrational triangle. -/ theorem irrational_triangle_invariant_set (Q : Triangle) (hQ : Q.HasIrrationalAngle) (A : Set Phase) (hA : MeasurableSet A) (hAi : ∀ t : ℝ, phaseMeasure Q (((billiardFlow Q t) ⁻¹' A) ∆ A)=0) : phaseMeasure Q A=0 ∨ phaseMeasure Q A=1 := by classical let f : Phase → ℂ := A.indicator (fun _ => 1) have hfm : StronglyMeasurable f := stronglyMeasurable_const.indicator hA have hf : MemLp f 2 (phaseMeasure Q) := (memLp_const (1:ℂ)).indicator hA have hb (z : Phase) : ‖f z‖ ≤ 1 := by by_cases hz : z∈A <;> simp [f,hz] obtain ⟨c,hc⟩ := bounded_billiard_invariant_constant Q hQ hfm hf hb (fun t => invariant_indicator _ _ _ (hAi t)) exact measure_zero_or_one_of_indicator_constant hc /-- Full two-sided billiard ergodicity, including complete regular chains and the geometric and measure-theoretic guarantees. -/ theorem irrational_triangle_billiard (Q : Triangle) (hQ : Q.HasIrrationalAngle) : MainConclusion Q := by refine ⟨phaseMeasure_univ Q,ae_exists_chain Q,?_,measurePreserving_billiardFlow Q, billiardFlow_cocycle Q,irrational_triangle_invariant_set Q hQ⟩ intro z c d exact FlightChain.unique c d /-- The same conclusion for the completion of the area-angular measure. Thus "measurable" may also be read in the Lebesgue-completed sense. -/ theorem irrational_triangle_nullMeasurable_set (Q : Triangle) (hQ : Q.HasIrrationalAngle) (A : Set Phase) (hA : NullMeasurableSet A (phaseMeasure Q)) (hAi : ∀ t : ℝ, phaseMeasure Q (((billiardFlow Q t) ⁻¹' A) ∆ A)=0) : phaseMeasure Q A=0 ∨ phaseMeasure Q A=1 := by have he : toMeasurable (phaseMeasure Q) A =ᵐ[phaseMeasure Q] A := hA.toMeasurable_ae_eq rw [← measure_congr he] apply irrational_triangle_invariant_set Q hQ _ (measurableSet_toMeasurable _ _) intro t apply measure_symmDiff_eq_zero_iff.mpr have hpre : (billiardFlow Q t ⁻¹' toMeasurable (phaseMeasure Q) A) =ᵐ[phaseMeasure Q] (billiardFlow Q t ⁻¹' A) := (measurePreserving_billiardFlow Q t).quasiMeasurePreserving.ae he exact hpre.trans ((measure_symmDiff_eq_zero_iff.mp (hAi t)).trans he.symm) end TriangularBilliards end end end end end end end end end end end end end OAI