File size: 2,944 Bytes
a44267b | 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 | import OAI.NumberTheory.DukePrimeDegree.Entropy.RootPrefixFutureFibers
namespace OAI
/-! # Positive leaf mass of countable whole-future atoms
The argument uses support of the actual conditional leaf kernel. No finite
entropy of the countable observation is required.
-/
noncomputable section
open MeasureTheory ProbabilityTheory Set Filter
open scoped Topology
namespace DukePrimeDegree
open PrimeDegreePackets
namespace RootPrefixCoordinates
variable {n : ℕ} {P₀ : Finset (Fin n × Fin n)} {D : Finset (Fin n)}
{V A : Type*} [TopologicalSpace V]
theorem wholeFutureAtom_mem_nhds (C : RootPrefixCoordinates n P₀ V)
(c : EliminationChain P₀ D) (hP₀ : InitialRootPattern P₀)
(hf : c.finalEntries = allPositiveRoots n) (g : G n)
(T : X n → X n) (Q : X n → A)
(z : (UpperRootTransverse n × V) × c.Time)
(hgood : ∀ᶠ v in 𝓝 (completeRootChain n).zeroTime, ∀ k : ℕ,
Q (T^[k] (rightAction (C.timeParam c hP₀ hf g z) ((completeRootChain n).word v))) =
Q (T^[k] (C.timeParam c hP₀ hf g z))) :
{t | ∀ k : ℕ, Q (T^[k] (C.timeParam c hP₀ hf g (z.1,t))) =
Q (T^[k] (C.timeParam c hP₀ hf g z))} ∈ 𝓝 z.2 := by
have hc : Tendsto (c.fullRootTime hP₀ hf ∘ c.remainingRebase hP₀ hf z.2)
(𝓝 z.2) (𝓝 (completeRootChain n).zeroTime) := by
have H := ((c.continuous_fullRootTime hP₀ hf).comp
(c.remainingRebase hP₀ hf z.2).continuous).tendsto z.2
simpa only [Function.comp_apply,c.remainingRebase_self,c.fullRootTime_zero] using H
filter_upwards [hc.eventually hgood] with t ht
intro k
rw [Function.comp_apply,c.word_fullRootTime] at ht
rw [C.timeParam_rebased c hP₀ hf g z.1 z.2 t]
exact ht k
end RootPrefixCoordinates
namespace RootPrefixBox
variable {n : ℕ} {P₀ : Finset (Fin n × Fin n)} {D : Finset (Fin n)}
{V A : Type*} [TopologicalSpace V] [SecondCountableTopology V]
[MeasurableSpace V] [BorelSpace V]
{C : RootPrefixCoordinates n P₀ V} {c : EliminationChain P₀ D}
{hP₀ : InitialRootPattern P₀} {hf : c.finalEntries = allPositiveRoots n} {g : G n}
omit [SecondCountableTopology V] in
theorem ae_wholeFutureAtom_pos (B : RootPrefixBox C c hP₀ hf g)
(μ : Measure (X n)) [IsFiniteMeasure μ] (T : X n → X n) (Q : X n → A)
(hgood : ∀ᵐ x ∂μ, ∀ᶠ v in 𝓝 (completeRootChain n).zeroTime, ∀ k : ℕ,
Q (T^[k] (rightAction x ((completeRootChain n).word v))) = Q (T^[k] x)) :
∀ᵐ z ∂B.coordinateMeasure μ, 0 < B.conditionalKernel μ z.1
{t | ∀ k : ℕ, Q (T^[k] (C.timeParam c hP₀ hf g (z.1,t))) =
Q (T^[k] (C.timeParam c hP₀ hf g z))} := by
filter_upwards [B.ae_coordinate_of_ambient μ hgood,B.ae_mem_conditionalKernel_support μ]
with z hz hs
apply ((B.conditionalKernel μ z.1).mem_support_iff_forall z.2).mp hs
exact C.wholeFutureAtom_mem_nhds c hP₀ hf g T Q z hz
end RootPrefixBox
end DukePrimeDegree
end
end OAI
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