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Uniqueness of the fast spaces

Lax606786.FastSpaceUniqueness · concepts/Lax606786/FastSpaceUniqueness.lean · lax-606786

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    Natural Language Statement

    Theorem

    Let L\mathcal{L} be a cocycle on a separable Banach space with its Borel σ\sigma-algebra, with distinct exponents almost surely equal to λi\lambda_i, and let i≥1i \ge 1. Let E,E′E, E' be measurable equivariant families of subspaces of dimensions almost surely equal to mi≥1m_i \ge 1, both growing at rate λi\lambda_i in norm and in slowest growth. Then E(ω)=E′(ω)E(\omega) = E'(\omega) almost surely.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
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    Lean source view on GitLab

    1import Lax606786.Cocycles
    2import Lax606786.BanachGrassmannian
    3
    4/-!
    5---
    6title: Uniqueness of the fast spaces
    7type: theorem
    8---
    9Let L\mathcal{L} be a cocycle on a separable Banach space with its Borel σ\sigma-algebra, with
    10distinct exponents almost surely equal to λi\lambda_i, and let i≥1i \ge 1. Let E,E′E, E' be measurable
    11equivariant families of subspaces of dimensions almost surely equal to mi≥1m_i \ge 1, both growing at
    12rate λi\lambda_i in norm and in slowest growth. Then E(ω)=E′(ω)E(\omega) = E'(\omega) almost surely.
    13-/
    14
    15namespace Lax606786.FastSpaceUniqueness
    16
    17open MeasureTheory Filter Topology TopologicalSpace
    18open Lax606786.Cocycles Lax606786.BanachGrassmannian Lax606786.OperatorStatistics
    19 Lax606786.ExtendedLog
    20
    21/-- Two fast families at the same exponent and multiplicity agree almost surely. -/
    22axiom fast_space_unique {Ω X : Type*} [MeasurableSpace Ω] [StandardBorelSpace Ω]
    23 [NormedAddCommGroup X] [NormedSpace ℝ X] [MeasurableSpace X]
    24 (R : Cocycle Ω X) [SeparableSpace X] [BorelSpace X]
    25 [CompleteSpace X] {lam : ℕ → EReal} (hlam : ∀ i, ∀ᵐ ω ∂(R.μ), R.lambda i ω = lam i)
    26 (i : ℕ) (hi : 1 ≤ i) {m m' : ℕ} [NeZero m] [NeZero m']
    27 (hm : ∀ᵐ ω ∂(R.μ), R.mult i ω = m) (hm' : ∀ᵐ ω ∂(R.μ), R.mult i ω = m')
    28 (E : Ω → GrassmannianFin X m) (E' : Ω → GrassmannianFin X m')
    29 (hEm : Measurable E) (hE'm : Measurable E')
    30 (hEeq : ∀ᵐ ω ∂(R.μ), Submodule.map (R.L ω : X →ₗ[ℝ] X) ((E ω).1 : Submodule ℝ X)
    31 = ((E (R.σ ω)).1 : Submodule ℝ X))
    32 (hE'eq : ∀ᵐ ω ∂(R.μ), Submodule.map (R.L ω : X →ₗ[ℝ] X) ((E' ω).1 : Submodule ℝ X)
    33 = ((E' (R.σ ω)).1 : Submodule ℝ X))
    34 (hEn : ∀ᵐ ω ∂(R.μ), Tendsto (fun n : ℕ => logEReal
    35 ‖(R.iterate n ω).comp (((E ω).1 : Submodule ℝ X)).subtypeL‖ / (n : EReal)) atTop
    36 (𝓝 (lam i)))
    37 (hE'n : ∀ᵐ ω ∂(R.μ), Tendsto (fun n : ℕ => logEReal
    38 ‖(R.iterate n ω).comp (((E' ω).1 : Submodule ℝ X)).subtypeL‖ / (n : EReal)) atTop
    39 (𝓝 (lam i)))
    40 (hEg : ∀ᵐ ω ∂(R.μ), Tendsto (fun n : ℕ => logEReal
    41 (growth (R.iterate n ω) ((E ω).1 : Submodule ℝ X)) / (n : EReal)) atTop (𝓝 (lam i)))
    42 (hE'g : ∀ᵐ ω ∂(R.μ), Tendsto (fun n : ℕ => logEReal
    43 (growth (R.iterate n ω) ((E' ω).1 : Submodule ℝ X)) / (n : EReal)) atTop
    44 (𝓝 (lam i))) :
    45 ∀ᵐ ω ∂(R.μ), ((E ω).1 : Submodule ℝ X) = ((E' ω).1 : Submodule ℝ X)
    46
    47end Lax606786.FastSpaceUniqueness
    48
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