| import OAI.NumberTheory.DukePrimeDegree.Entropy.AtomicFutureCover |
| import OAI.NumberTheory.DukePrimeDegree.Entropy.SmallFutureObservation |
| import OAI.NumberTheory.DukePrimeDegree.Entropy.CountablePatchEntropy |
| import OAI.NumberTheory.DukePrimeDegree.Entropy.PredecessorFutureInformation |
|
|
| namespace OAI |
|
|
| /-! # Atomic expanding roots force zero metric entropy -/ |
|
|
| noncomputable section |
| open MeasureTheory Set TopologicalSpace |
| open scoped Topology |
| namespace DukePrimeDegree |
| open PrimeDegreePackets |
|
|
| theorem diagonal_entropy_zero_of_atomic_positive_roots (d : ℕ) |
| (ell : Fin (d+1) → ℝ) (hell : ∑ i, ell i=0) |
| (hgap : ∀ p : allPositiveRoots (d+1), Real.log 4 ≤ ell p.val.2-ell p.val.1) |
| (μ : Measure (X (d+1))) [IsProbabilityMeasure μ] |
| [ErgodicSMul (Multiplicative (LogCoordinates d)) (X (d+1)) μ] |
| (hatom : ∀ p : ↥(allPositiveRoots (d+1) \ ∅), ∀ᵐ x ∂μ, |
| intrinsicRootLeaf (ne_of_lt (Finset.mem_filter.mp (Finset.mem_sdiff.mp p.prop).1).2) μ x = |
| Measure.dirac 0) : |
| measureEntropy μ (fun x => rightAction x (diagonalOneParameter ell hell (-1)))=0 := by |
| have hA (u : LogCoordinates d) : μ.map (fun x => rightAction x (diagonalFlow u))=μ := |
| (measurePreserving_smul (Multiplicative.ofAdd u) μ).map_eq |
| have hA' (v : transverseNeutralSpace (d+1)) : |
| μ.map (fun x => rightAction x (diagonalOneParameter v.val v.prop 1))=μ := by |
| rw [diagonalOneParameter_eq_diagonalFlow] |
| exact hA _ |
| have hord : StrictMono ell := by |
| intro i j hij |
| have hh := hgap ⟨(i,j),Finset.mem_filter.mpr ⟨Finset.mem_univ _,hij⟩⟩ |
| have hp : 0 < Real.log 4 := Real.log_pos (by norm_num) |
| dsimp at hh |
| linarith |
| obtain ⟨H,Q,r,hH,hQ,hanti,hr,hinj,hbranch,hlift,hbilateral⟩ := |
| exists_small_future_observation d ell hell hgap μ hA |
| obtain ⟨s,hsm,hs⟩ := exists_measurable_quotient_section (d+1) |
| let K₀ : PositiveCompacts (transverseNeutralSpace (d+1)) := Classical.arbitrary _ |
| let η := Measure.addHaarMeasure K₀ |
| let z₀ : NeutralCoordinateBase (d+1) × transverseNeutralSpace (d+1) := |
| ((⟨upperRootIdentity (d+1),by intro i; simp [upperRootIdentity]⟩, |
| (completeRootChain (d+1)).zeroTime),0) |
| obtain ⟨E,hE,hfull,hpair⟩ := exists_atomic_neutral_future_cover d ell hell hord Q H hH |
| r hanti hr hinj hlift μ s hsm hs hA' η K₀ K₀.isCompact Measure.addHaarMeasure_self |
| z₀ hatom |
| let T := fun x => rightAction x (diagonalOneParameter ell hell (-1)) |
| have hT : MeasurePreserving T μ μ := by |
| refine ⟨(continuous_rightAction _).measurable,?_⟩ |
| dsimp only [T] |
| rw [diagonalOneParameter_eq_diagonalFlow] |
| exact hA _ |
| obtain ⟨M,B,decode,hB,href,hdecode,hupper⟩ := |
| finite_predecessors_entropy_upper μ hT Q hQ hbranch |
| have hzero := predecessor_information_zero_of_sublinear_words μ hT Q hQ B hB href |
| decode hdecode (fun ε hε => |
| countable_patch_name_entropy_sublinear ell hell Q hQ E hE hpair μ hfull hT B hB hε) |
| apply le_antisymm _ bot_le |
| have hbound := measureEntropy_le_of_bilateral_heightClip_bounds d ell hell Q hQ μ hA |
| (fun x y hxy => hbilateral x y hxy.1 hxy.2) 0 |
| (fun L => by simpa only [hzero] using hupper L) |
| simpa only [ENNReal.ofReal_zero,bot_eq_zero] using hbound |
|
|
| end DukePrimeDegree |
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|
| end |
|
|
| end OAI |
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