openAI-math-dataset / lean /OAI /NumberTheory /DukePrimeDegree /Geometry /AtomlessDiagonalMeasure.lean
| import OAI.NumberTheory.DukePrimeDegree.Dynamics.SmallCentralizer | |
| import OAI.NumberTheory.DukePrimeDegree.Entropy.CompactFutureLifts | |
| import OAI.NumberTheory.DukePrimeDegree.Dynamics.DiagonalFlowMaximal | |
| import Mathlib.Topology.Separation.Lemmas | |
| namespace OAI | |
| /-! # Diagonal-invariant probability measures have no point atoms -/ | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped Topology | |
| namespace DukePrimeDegree | |
| open PrimeDegreePackets | |
| lemma measure_singleton_eq_zero_of_continuous_orbit | |
| {Y : Type*} [MetricSpace Y] [MeasurableSpace Y] [MeasurableSingletonClass Y] | |
| (μ : Measure Y) [IsFiniteMeasure μ] (x : Y) (f : ℝ → Y) | |
| (hf : Continuous f) (hf0 : f 0=x) | |
| (hmass : ∀ t, μ {f t}=μ {x}) (hne : ∃ t, f t ≠ x) : μ {x}=0 := by | |
| by_contra hx | |
| have hc : {y : Y | 0 < μ {y}}.Countable := by | |
| simpa only [id_eq,Set.ofPred_eq_eq_singleton] using | |
| (Measure.countable_meas_level_set_pos (μ := μ) (g := id) measurable_id) | |
| have hr : (range f).Countable := hc.mono (by | |
| rintro _ ⟨t,rfl⟩ | |
| change 0 < μ {f t} | |
| rw [hmass] | |
| exact pos_iff_ne_zero.mpr hx) | |
| have hp : IsPreconnected (range f) := isPreconnected_range hf | |
| have hs : (range f).Subsingleton := hr.isTotallyDisconnected _ (subset_refl _) hp | |
| obtain ⟨t,ht⟩ := hne | |
| exact ht ((hs ⟨t,rfl⟩ ⟨0,rfl⟩).trans hf0) | |
| /-- Only one nontrivial one-parameter diagonal flow is needed. -/ | |
| theorem nullSingletonClass_of_diagonal_invariant (d : ℕ) | |
| {i j : Fin (d+1)} (hij : i ≠ j) | |
| (μ : Measure (X (d+1))) [IsProbabilityMeasure μ] | |
| (hA : ∀ u : LogCoordinates d, μ.map (fun x => rightAction x (diagonalFlow u))=μ) : | |
| NullSingletonClass μ := by | |
| let : MetricSpace (X (d+1)) := TopologicalSpace.metrizableSpaceMetric (X (d+1)) | |
| let ell := rootContractionWeights i j | |
| let D := diagonalOneParameter ell (rootContractionWeights_sum i j) | |
| have hD : Continuous D := continuous_diagonalOneParameter ell _ | |
| constructor | |
| intro x | |
| apply measure_singleton_eq_zero_of_continuous_orbit μ x (fun t => rightAction x (D t)) | |
| ((continuous_rightAction_uncurry _).comp (continuous_const.prodMk hD)) | |
| (by simp only [D,diagonalOneParameter_zero,rightAction_one]) | |
| · intro t | |
| have hpres : MeasurePreserving (fun y => rightAction y (D t)) μ μ := by | |
| refine ⟨(continuous_rightAction _).measurable,?_⟩ | |
| dsimp only [D] | |
| rw [diagonalOneParameter_eq_diagonalFlow] | |
| exact hA _ | |
| have h := hpres.measure_preimage (measurableSet_singleton (rightAction x (D t))).nullMeasurableSet | |
| have he : (fun y => rightAction y (D t)) ⁻¹' {rightAction x (D t)}={x} := by | |
| ext y | |
| exact (rightActionHomeomorph (D t)).injective.eq_iff | |
| rw [he] at h | |
| exact h.symm | |
| · obtain ⟨r,hr,hinj⟩ := compact_uniform_chart_injective ({x} : Set (X (d+1))) isCompact_singleton | |
| have hnear : {t : ℝ | D t ∈ matrixBall (d+1) r} ∈ 𝓝 0 := | |
| hD.continuousAt.preimage_mem_nhds | |
| ((matrixBall_isOpen _ _).mem_nhds (by simpa only [D,diagonalOneParameter_zero] using | |
| (one_mem_matrixBall (d+1) hr))) | |
| obtain ⟨δ,hδ,hsub⟩ := Metric.mem_nhds_iff.mp hnear | |
| have ht : 0 < δ/2 := by positivity | |
| have htm : D (δ/2) ∈ matrixBall (d+1) r := hsub (by | |
| simpa only [Metric.mem_ball,Real.dist_eq,sub_zero,abs_of_pos ht] using | |
| (show δ/2 < δ by linarith)) | |
| refine ⟨δ/2,?_⟩ | |
| intro heq | |
| have he : D (δ/2)=1 := hinj x (mem_singleton x) htm | |
| (one_mem_matrixBall (d+1) hr) (by simpa only [rightAction_one] using heq) | |
| have hei := congrArg (fun g : G (d+1) => g.val i i) he | |
| have hexp : Real.exp (δ/2)=1 := by | |
| simpa [D,ell,diagonalOneParameter,rootContractionWeights,hij] using hei | |
| have hz : δ/2=0 := Real.exp_injective (hexp.trans Real.exp_zero.symm) | |
| linarith | |
| end DukePrimeDegree | |
| end | |
| end OAI | |