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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 | import OAI.Dynamics.StandardMap.Lyapunov.Definitions
namespace OAI
section ComponentsStatement
open MeasureTheory Set Filter
open scoped ENNReal Topology
namespace StandardMapEntropy
noncomputable def normalizedArea (E : Set Torus) : Measure Torus :=
(area E)⁻¹ • area.restrict E
def IsBernoulli (f : Torus → Torus) (μ : Measure Torus) : Prop :=
∃ (β : Measure ℝ), IsProbabilityMeasure β ∧
∃ (encode : Torus → (ℤ → ℝ)) (decode : (ℤ → ℝ) → Torus),
MeasurePreserving encode μ (MeasureTheory.Measure.infinitePi (fun _ : ℤ => β)) ∧
MeasurePreserving decode (MeasureTheory.Measure.infinitePi (fun _ : ℤ => β)) μ ∧
(∀ᵐ z ∂μ, decode (encode z) = z) ∧
(∀ᵐ w ∂MeasureTheory.Measure.infinitePi (fun _ : ℤ => β), encode (decode w) = w) ∧
(∀ᵐ z ∂μ, encode (f z) = fun i : ℤ => encode z (i+1))
def HyperbolicBernoulliComponent (k : ℝ) : Prop :=
∃ E : Set Torus, MeasurableSet E ∧ 0 < area E ∧
((standardMap k) ⁻¹' E =ᵐ[area] E) ∧
Ergodic (standardMap k) (normalizedArea E) ∧
(∀ᵐ z ∂normalizedArea E, ∃ l : ℝ, 0 < l ∧ LyapunovSpectrumAt k z l) ∧
∃ (N : ℕ) (hN : 0 < N) (P : Fin N → Set Torus),
(∀ j, MeasurableSet (P j)) ∧
((⋃ j, P j) =ᵐ[area] E) ∧
(∀ i j, i ≠ j → area (P i ∩ P j) = 0) ∧
(∀ j : Fin N, (standardMap k '' P j) =ᵐ[area]
P ⟨(j.val+1)%N, Nat.mod_lt _ hN⟩) ∧
(∀ j, area (P j) = area E / (N : ℝ≥0∞) ∧ 0 < area (P j)) ∧
(∀ j, IsBernoulli ((standardMap k)^[N]) (normalizedArea (P j)))
def MainComponentsObligation : Prop :=
∃ k₀ : ℝ, 0 < k₀ ∧ ∀ k : ℝ, k₀ ≤ k → HyperbolicBernoulliComponent k
end StandardMapEntropy
end ComponentsStatement
end OAI
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