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The Lyapunov exponents are limits, forwards, backwards and over balanced intervals

Lax606786.ExponentLimits · concepts/Lax606786/ExponentLimits.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 k≥1k \ge 1. Then χk\chi_k is measurable, and for almost every ω\omega the lim sup⁡\limsup defining it is a limit, along forward orbits, along backward orbits and over balanced intervals:

    χk(ω)=lim⁡n1nlog⁡ρk(Lω(n))=lim⁡n1nlog⁡ρk(Lσ−nω(n))=lim⁡n12nlog⁡ρk(Lσ−nω(2n)).\chi_k(\omega) = \lim_n \tfrac1n \log \rho_k(\mathcal{L}^{(n)}_\omega) = \lim_n \tfrac1n \log \rho_k(\mathcal{L}^{(n)}_{\sigma^{-n}\omega}) = \lim_n \tfrac1{2n} \log \rho_k(\mathcal{L}^{(2n)}_{\sigma^{-n}\omega}).

    Moreover χ1≥χ2≥⋯\chi_1 \ge \chi_2 \ge \cdots pointwise, ν≤χk\nu \le \chi_k, and almost everywhere χ1≤∫log⁡+∥L∥ dμ<∞\chi_1 \le \int \log^+ \|\mathcal{L}\| \, d\mu < \infty.

    The χk\chi_k also decrease to the Bernstein index ν=inf⁡kχk\nu = \inf_k \chi_k at every point.

    Concept map
    6 concepts
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 9 statements. Each proof establishes one of them relative to its assumptions.

    1 ae_chi_one_le_integral proven

    2 ae_chi_one_lt_top proven

    3 ae_tendsto_chi proven

    4 ae_tendsto_chi_backward proven

    5 ae_tendsto_chi_balanced proven

    6 chi_antitone proven

    7 measurable_chi proven

    8 nu_le_chi proven

    9 tendsto_chi_nu proven

    Lean source view on GitLab

    1import Lax606786.Cocycles
    2import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
    3import Mathlib.MeasureTheory.Integral.Bochner.Basic
    4
    5/-!
    6---
    7title: The Lyapunov exponents are limits, forwards, backwards and over balanced intervals
    8type: theorem
    9---
    10Let L\mathcal{L} be a cocycle on a separable Banach space XX with its Borel σ\sigma-algebra, and
    11k≥1k \ge 1. Then χk\chi_k is measurable, and for almost every ω\omega the lim sup⁡\limsup defining it is a
    12limit, along forward orbits, along backward orbits and over balanced intervals:
    13χk(ω)=lim⁡n1nlog⁡ρk(Lω(n))=lim⁡n1nlog⁡ρk(Lσ−nω(n))=lim⁡n12nlog⁡ρk(Lσ−nω(2n)).\chi_k(\omega) = \lim_n \tfrac1n \log \rho_k(\mathcal{L}^{(n)}_\omega) = \lim_n \tfrac1n \log \rho_k(\mathcal{L}^{(n)}_{\sigma^{-n}\omega}) = \lim_n \tfrac1{2n} \log \rho_k(\mathcal{L}^{(2n)}_{\sigma^{-n}\omega}).
    14
    15
    16Moreover χ1≥χ2≥⋯\chi_1 \ge \chi_2 \ge \cdots pointwise, ν≤χk\nu \le \chi_k, and almost everywhere
    17χ1≤∫log⁡+∥L∥ dμ<∞\chi_1 \le \int \log^+ \|\mathcal{L}\| \, d\mu < \infty.
    18
    19The χk\chi_k also decrease to the Bernstein index ν=inf⁡kχk\nu = \inf_k \chi_k at every point.
    20-/
    21
    22namespace Lax606786.ExponentLimits
    23
    24open MeasureTheory Filter Topology TopologicalSpace
    25open Lax606786.Cocycles Lax606786.OperatorStatistics Lax606786.ExtendedLog
    26
    27variable {Ω X : Type*} [MeasurableSpace Ω] [StandardBorelSpace Ω]
    28 [NormedAddCommGroup X] [NormedSpace ℝ X] [MeasurableSpace X]
    29
    30/-- `χ_k` is measurable. -/
    31axiom measurable_chi (R : Cocycle Ω X) [SeparableSpace X] [BorelSpace X] (k : ℕ) [NeZero k] :
    32 Measurable (R.chi k)
    33
    34/-- `χ_k` is an a.e. limit along forward orbits. -/
    35axiom ae_tendsto_chi (R : Cocycle Ω X) [SeparableSpace X] [BorelSpace X] (k : ℕ) [NeZero k] :
    36 ∀ᵐ ω ∂(R.μ), Tendsto (fun n : ℕ =>
    37 logEReal (bernsteinNumber (R.iterate n ω) k) / (n : EReal)) atTop (𝓝 (R.chi k ω))
    38
    39/-- `χ_k` is an a.e. limit along backward orbits. -/
    40axiom ae_tendsto_chi_backward (R : Cocycle Ω X) [SeparableSpace X] [BorelSpace X] (k : ℕ)
    41 [NeZero k] :
    42 ∀ᵐ ω ∂(R.μ), Tendsto (fun n : ℕ =>
    43 logEReal (bernsteinNumber (R.iterate n ((R.σ.symm : Ω → Ω)^[n] ω)) k) / (n : EReal))
    44 atTop (𝓝 (R.chi k ω))
    45
    46/-- `χ_k` is an a.e. limit over the balanced intervals `[-n, n)`. -/
    47axiom ae_tendsto_chi_balanced (R : Cocycle Ω X) [SeparableSpace X] [BorelSpace X] (k : ℕ)
    48 [NeZero k] :
    49 ∀ᵐ ω ∂(R.μ), Tendsto (fun n : ℕ => logEReal
    50 (bernsteinNumber (R.iterate (2 * n) ((R.σ.symm : Ω → Ω)^[n] ω)) k) /
    51 ((2 * n : ℕ) : EReal))
    52 atTop (𝓝 (R.chi k ω))
    53
    54/-- `χ_1 ≥ χ_2 ≥ ⋯`. -/
    55axiom chi_antitone (R : Cocycle Ω X) (ω : Ω) : Antitone fun k : ℕ => R.chi (k + 1) ω
    56
    57/-- `ν ≤ χ_k` for `k ≥ 1`. -/
    58axiom nu_le_chi (R : Cocycle Ω X) (ω : Ω) {k : ℕ} (hk : 1 ≤ k) : R.nu ω ≤ R.chi k ω
    59
    60/-- `χ_1 < ∞` a.e. -/
    61axiom ae_chi_one_lt_top (R : Cocycle Ω X) [SeparableSpace X] [BorelSpace X] :
    62 ∀ᵐ ω ∂(R.μ), R.chi 1 ω < ⊤
    63
    64/-- `χ_1 ≤ ∫ log⁺‖𝓛‖` a.e. -/
    65axiom ae_chi_one_le_integral (R : Cocycle Ω X) :
    66 ∀ᵐ ω ∂(R.μ), R.chi 1 ω ≤ ((∫ ω, Real.log (max ‖R.L ω‖ 1) ∂(R.μ) : ℝ) : EReal)
    67
    68/-- The `χ_k` decrease to `ν`. -/
    69axiom tendsto_chi_nu (R : Cocycle Ω X) (ω : Ω) :
    70 Tendsto (fun k : ℕ => R.chi (k + 1) ω) atTop (𝓝 (R.nu ω))
    71
    72end Lax606786.ExponentLimits
    73
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