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Uniqueness of fast families below a spectral gap

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

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    Natural Language Statement

    Theorem

    Let L\mathcal{L} be a cocycle on a separable Banach space XX with its Borel σ\sigma-algebra, and suppose χm+1<b<a\chi_{m+1} < b < a almost surely. Two measurable equivariant families E,E′ ⁣:Ω→GmXE, E' \colon \Omega \to \mathcal{G}_m X on which Lω(n)\mathcal{L}^{(n)}_\omega eventually grows faster than eane^{an} agree almost everywhere.

    This generalises the uniqueness clause of Lemma 28 of Lee (2024).

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    1import Lax606786.Cocycles
    2import Lax606786.BanachGrassmannian
    3
    4/-!
    5---
    6title: Uniqueness of fast families below a spectral gap
    7type: theorem
    8---
    9Let L\mathcal{L} be a cocycle on a separable Banach space XX with its Borel σ\sigma-algebra, and
    10suppose χm+1<b<a\chi_{m+1} < b < a almost surely. Two measurable equivariant families
    11E,E′ ⁣:Ω→GmXE, E' \colon \Omega \to \mathcal{G}_m X on which Lω(n)\mathcal{L}^{(n)}_\omega eventually grows
    12faster than eane^{an} agree almost everywhere.
    13
    14This generalises the uniqueness clause of Lemma 28 of Lee (2024).
    15-/
    16
    17namespace Lax606786.GapUniqueness
    18
    19open MeasureTheory Filter TopologicalSpace
    20open Lax606786.Cocycles Lax606786.BanachGrassmannian Lax606786.OperatorStatistics
    21
    22variable {Ω X : Type*} [MeasurableSpace Ω] [StandardBorelSpace Ω]
    23 [NormedAddCommGroup X] [NormedSpace ℝ X] [MeasurableSpace X]
    24
    25/-- Two equivariant families growing faster than `e^{an}`, with `a` above `χ_{m+1}`, agree. -/
    26axiom ae_eq_of_fast (R : Cocycle Ω X) [SeparableSpace X] [BorelSpace X]
    27 (m : ℕ) [NeZero m] {a b : ℝ} (hab : b < a)
    28 (hchi2 : ∀ᵐ ω ∂(R.μ), R.chi (m + 1) ω < (b : EReal))
    29 (E E' : Ω → GrassmannianFin X m) (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 (hEfast : ∀ᵐ ω ∂(R.μ), ∀ᶠ n : ℕ in atTop,
    35 Real.exp (a * n) < growth (R.iterate n ω) ((E ω).1 : Submodule ℝ X))
    36 (hE'fast : ∀ᵐ ω ∂(R.μ), ∀ᶠ n : ℕ in atTop,
    37 Real.exp (a * n) < growth (R.iterate n ω) ((E' ω).1 : Submodule ℝ X)) :
    38 ∀ᵐ ω ∂(R.μ), E ω = E' ω
    39
    40end Lax606786.GapUniqueness
    41
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