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The slow spaces grow uniformly at their exponent

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

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    Natural Language Statement

    Theorem

    Let L\mathcal{L} be a cocycle on a separable Banach space and (L,λ,m,E,V,Π)(L, \lambda, m, E, V, \Pi) an Oseledets decomposition of it. Then for 2≤l≤L2 \le l \le L, almost every ω\omega and every real b>λlb > \lambda_l, eventually

    ∥Lω(n)v∥≤ebn∥v∥for all v∈Vl(ω),\|\mathcal{L}^{(n)}_\omega v\| \le e^{bn}\|v\| \quad \text{for all } v \in V_l(\omega),

    that is, lim sup⁡n1nlog⁡∥Lω(n)∣Vl(ω)∥≤λl\limsup_n \frac1n \log \|\mathcal{L}^{(n)}_\omega|_{V_l(\omega)}\| \le \lambda_l.

    The decomposition only records that each vector of Vl(ω)V_l(\omega) grows at rate at most λl\lambda_l; this is the uniform version, which the construction of the decomposition provides (the slow estimate accompanying the top splitting) and which also follows from χd+1=λl\chi_{d+1} = \lambda_l, d=m1+⋯+ml−1d = m_1 + \cdots + m_{l-1}, by comparing ∥L(n)∣Vl∥\|\mathcal{L}^{(n)}|_{V_l}\| with the Bernstein number ρd+1(L(n))\rho_{d+1}(\mathcal{L}^{(n)}) of E1⊕⋯⊕El−1⊕RvE_1 \oplus \cdots \oplus E_{l-1} \oplus \mathbb{R}v.

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    Lean source view on GitLab

    1import Lax606786.OseledetsDecompositions
    2
    3/-!
    4---
    5title: The slow spaces grow uniformly at their exponent
    6type: theorem
    7---
    8Let L\mathcal{L} be a cocycle on a separable Banach space and (L,λ,m,E,V,Π)(L, \lambda, m, E, V, \Pi)
    9an Oseledets decomposition of it. Then for 2≤l≤L2 \le l \le L, almost every ω\omega and every
    10real b>λlb > \lambda_l, eventually
    11∥Lω(n)v∥≤ebn∥v∥for all v∈Vl(ω),\|\mathcal{L}^{(n)}_\omega v\| \le e^{bn}\|v\| \quad \text{for all } v \in V_l(\omega),
    12that is, lim sup⁡n1nlog⁡∥Lω(n)∣Vl(ω)∥≤λl\limsup_n \frac1n \log \|\mathcal{L}^{(n)}_\omega|_{V_l(\omega)}\| \le \lambda_l.
    13
    14The decomposition only records that each vector of Vl(ω)V_l(\omega) grows at rate at most λl\lambda_l;
    15this is the uniform version, which the construction of the decomposition provides (the slow estimate
    16accompanying the top splitting) and which also follows from χd+1=λl\chi_{d+1} = \lambda_l, d=m1+⋯+ml−1d = m_1 + \cdots + m_{l-1}
    17, by comparing ∥L(n)∣Vl∥\|\mathcal{L}^{(n)}|_{V_l}\| with the Bernstein number
    18ρd+1(L(n))\rho_{d+1}(\mathcal{L}^{(n)}) of E1⊕⋯⊕El−1⊕RvE_1 \oplus \cdots \oplus E_{l-1} \oplus \mathbb{R}v.
    19-/
    20
    21namespace Lax606786.SlowSpaceUniformGrowth
    22
    23open MeasureTheory Filter TopologicalSpace
    24open Lax606786.BanachGrassmannian Lax606786.Cocycles Lax606786.OseledetsDecompositions
    25
    26/-- Uniform forward growth bound on the slow spaces. -/
    27axiom forward_norm_slowSpace_le {Ω X : Type*} [MeasurableSpace Ω] [StandardBorelSpace Ω]
    28 [NormedAddCommGroup X] [NormedSpace ℝ X] [MeasurableSpace X]
    29 [SeparableSpace X] [BorelSpace X] [CompleteSpace X]
    30 {R : Cocycle Ω X} {L : ℕ∞} {lam : ℕ → EReal}
    31 {m : ℕ → ℕ+} {E : ∀ i : ℕ, 1 ≤ i → (i : ℕ∞) < L → Ω → GrassmannianFin X (m i)}
    32 {V : ℕ → Ω → Submodule ℝ X} {P : ℕ → Ω → X →L[ℝ] X}
    33 (h : IsOseledetsDecomposition R L lam m E V P)
    34 {l : ℕ} (hl₁ : 2 ≤ l) (hl₂ : (l : ℕ∞) ≤ L) :
    35 ∀ᵐ ω ∂(R.μ), ∀ b : ℝ, lam l < (b : EReal) → ∀ᶠ n : ℕ in atTop,
    36 ∀ v ∈ V l ω, ‖R.iterate n ω v‖ ≤ Real.exp (b * n) * ‖v‖
    37
    38end Lax606786.SlowSpaceUniformGrowth
    39
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