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Uniqueness of the Oseledets decomposition

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

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    Natural Language Statement

    Theorem

    Let L\mathcal{L} be a cocycle on a separable Banach space XX. Any two Oseledets decompositions (L,λ,m,E,V,Π)(L, \lambda, m, E, V, \Pi) and (L′,λ′,m′,E′,V′,Π′)(L', \lambda', m', E', V', \Pi') of L\mathcal{L} agree. They have L=L′L = L', λi=λi′\lambda_i = \lambda'_i for i≤Li \le L, and mi=mi′m_i = m'_i for i<Li < L. For every i<Li < L and almost every ω\omega,

    Ei(ω)=Ei′(ω),Vi+1(ω)=Vi+1′(ω),Πi+1(ω)=Πi+1′(ω).E_i(\omega) = E'_i(\omega), \qquad V_{i+1}(\omega) = V'_{i+1}(\omega), \qquad \Pi_{i+1}(\omega) = \Pi'_{i+1}(\omega).

    Nothing is asserted beyond these indices, where the decomposition theorem constrains nothing.

    Indices in the Lean statement are zero-based, as in the definition of Oseledets decompositions.

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    Lean source view on GitLab

    1import Lax606786.OseledetsDecompositions
    2
    3/-!
    4---
    5title: Uniqueness of the Oseledets decomposition
    6type: theorem
    7---
    8Let L\mathcal{L} be a cocycle on a separable Banach space XX.
    9Any two Oseledets decompositions (L,λ,m,E,V,Π)(L, \lambda, m, E, V, \Pi) and
    10(L′,λ′,m′,E′,V′,Π′)(L', \lambda', m', E', V', \Pi') of L\mathcal{L} agree. They have L=L′L = L',
    11λi=λi′\lambda_i = \lambda'_i for i≤Li \le L, and mi=mi′m_i = m'_i for i<Li < L. For every i<Li < L and
    12almost every ω\omega,
    13Ei(ω)=Ei′(ω),Vi+1(ω)=Vi+1′(ω),Πi+1(ω)=Πi+1′(ω).E_i(\omega) = E'_i(\omega), \qquad V_{i+1}(\omega) = V'_{i+1}(\omega), \qquad \Pi_{i+1}(\omega) = \Pi'_{i+1}(\omega).
    14
    15Nothing is asserted beyond these indices, where the decomposition theorem constrains nothing.
    16
    17Indices in the Lean statement are zero-based, as in the definition of Oseledets decompositions.
    18-/
    19
    20namespace Lax606786.OseledetsUniqueness
    21
    22open MeasureTheory Filter TopologicalSpace
    23open Lax606786.Grassmannian Lax606786.OperatorStatistics Lax606786.ExtendedLog
    24 Lax606786.Cocycles Lax606786.OseledetsDecompositions
    25
    26/-- Any two Oseledets decompositions of a cocycle agree, on every index the decomposition
    27theorem constrains. -/
    28axiom isOseledetsDecomposition_unique {Ω X : Type*} [MeasurableSpace Ω] [StandardBorelSpace Ω]
    29 [NormedAddCommGroup X] [NormedSpace ℝ X] [MeasurableSpace X]
    30 (R : Cocycle Ω X) [SeparableSpace X] [BorelSpace X] [CompleteSpace X]
    31 {Lval Lval' : ℕ∞} {lam lam' : ℕ → EReal} {mdim mdim' : ℕ → ℕ}
    32 {E : ∀ i : ℕ, ((i + 2 : ℕ) : ℕ∞) ≤ Lval → Ω → GrassmannianFin X (mdim i + 1)}
    33 {E' : ∀ i : ℕ, ((i + 2 : ℕ) : ℕ∞) ≤ Lval' → Ω → GrassmannianFin X (mdim' i + 1)}
    34 {V V' : ℕ → Ω → Submodule ℝ X} {P P' : ℕ → Ω → X →L[ℝ] X}
    35 (h : IsOseledetsDecomposition R Lval lam mdim E V P)
    36 (h' : IsOseledetsDecomposition R Lval' lam' mdim' E' V' P') :
    37 Lval = Lval' ∧ (∀ i : ℕ, (i : ℕ∞) < Lval → lam i = lam' i) ∧
    38 ∀ (i : ℕ) (hi : ((i + 2 : ℕ) : ℕ∞) ≤ Lval) (hi' : ((i + 2 : ℕ) : ℕ∞) ≤ Lval'),
    39 mdim i = mdim' i ∧ ∀ᵐ ω ∂(R.μ),
    40 ((E i hi ω).1 : Submodule ℝ X) = ((E' i hi' ω).1 : Submodule ℝ X) ∧
    41 V (i + 1) ω = V' (i + 1) ω ∧ P (i + 1) ω = P' (i + 1) ω
    42
    43end Lax606786.OseledetsUniqueness
    44
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