The Oseledets Multiplicative Ergodic Theorem

What the theorem precisely claims

Let (X, ℬ, μ, T) be a measure-preserving system with T ergodic, and let A: X → GL(d, ℝ) be measurable. Define the cocycle Aⁿ(x) = A(Tⁿ⁻¹x)⋯A(Tx)A(x), with A⁰(x) = I, so that Aⁿ⁺ᵐ(x) = Aⁿ(Tᵐx)Aᵐ(x). Assume the integrability condition log⁺‖A‖ ∈ L¹(μ), where log⁺ = max(log, 0).

Then there exist constants ∞ > λ₁ > λ₂ > ⋯ > λ_k ≥ −∞ (the Lyapunov exponents) with multiplicities m₁,…,m_k summing to d, and for μ-almost every x a measurable splitting ℝᵈ = E₁(x) ⊕ ⋯ ⊕ E_k(x), such that:

• dim Eᵢ(x) = mᵢ and A(x)Eᵢ(x) = Eᵢ(Tx) (equivariance);
• for every nonzero v ∈ Eᵢ(x), limₙ (1/n) log‖Aⁿ(x)v‖ = λᵢ.

In the invertible case (T invertible and log⁺‖A⁻¹‖ ∈ L¹) the same rate governs the past: (1/n) log‖A⁻ⁿ(x)v‖ → −λᵢ. The regularity is uniform in a strong sense — the sum of exponents equals limₙ (1/n) log|det Aⁿ(x)| = ∫ log|det A| dμ.

The picture: eigenvalues for non-commuting products

If all the A(x) were the same symmetric matrix A, the answer is elementary: (1/n) log‖Aⁿv‖ → log|η| where η is the eigenvalue of the largest eigenspace meeting v, and ℝᵈ splits into eigenspaces. Oseledets is the statement that this survives when the matrix changes at every step and the matrices do not commute — provided the changes are driven by an ergodic system so that time-averages exist.

The right object is not the eigenvalues of Aⁿ(x) but its singular values σ₁ⁿ ≥ ⋯ ≥ σ_dⁿ, the semi-axes of the ellipsoid Aⁿ(x)(unit ball). The theorem says each axis grows at a definite exponential rate: (1/n) log σⱼⁿ(x) → λ (the value with the right multiplicity). Geometrically, the image of the unit ball becomes an enormously eccentric ellipsoid whose axis lengths separate into k exponential scales, and Eᵢ(x) records the pre-image directions that land in the λᵢ-scale. It is a spectral theorem stretched along the time axis.

Key idea of the proof: subadditivity meets singular values

The engine is Kingman's subadditive ergodic theorem. The sequence fₙ(x) = log‖Aⁿ(x)‖ is subadditive: fₙ₊ₘ(x) ≤ fₙ(Tᵐx) + fₘ(x) because ‖Aⁿ⁺ᵐ‖ ≤ ‖Aⁿ∘Tᵐ‖‖Aᵐ‖. Integrability of log⁺‖A‖ gives f₁ ∈ L¹, so Kingman yields limₙ (1/n) log‖Aⁿ(x)‖ = λ₁ a.e. That is Furstenberg–Kesten: the top exponent exists. Applying the same argument to the exterior powers Λᵖ Aⁿ (a cocycle of the same type) gives limₙ (1/n) log‖Λᵖ Aⁿ‖ = λ₁ + ⋯ + λₚ, which recovers all the exponents as successive differences.

To get the splitting, form the symmetric positive-definite matrices (Aⁿ(x)ᵀAⁿ(x))^{1/(2n)}. Oseledets' key lemma: this converges a.e. to a limit Λ(x), whose eigenspaces are the Oseledets subspaces and whose log-eigenvalues are the λᵢ. Convergence follows from control of the singular values (via the exterior-power exponents) plus a Borel–Cantelli angle estimate showing the eigendirections stabilize. In the invertible case one intersects the forward filtration with the backward filtration (built from A⁻¹) to turn nested flags into a genuine direct sum.

Canonical example: the random Fibonacci / SL(2,ℝ) cocycle

Take X = {M, N}^ℤ with a Bernoulli(½,½) measure, T the shift, and A(x) = x₀ ∈ {M, N} ⊂ SL(2, ℝ). Since det = 1, the exponents satisfy λ₁ + λ₂ = ∫ log|det A| dμ = 0, so λ₂ = −λ₁ and the whole spectrum is {λ₁, −λ₁}. Furstenberg's theorem guarantees λ₁ > 0 (strict positivity) whenever the group generated by M, N is non-compact and strongly irreducible — the generic case. The Oseledets splitting ℝ² = E₁(x) ⊕ E₂(x) is then one expanding and one contracting line, varying measurably with the future/past of x.

A famous instance: the random Fibonacci sequence tₙ = ±tₙ₋₁ ± tₙ₋₂ with random signs. Its terms grow like eᵘⁿ where u = 0.1239755981… is Viswanath's constant (1999) — a Lyapunov exponent computed to high precision, an Oseledets exponent made concrete. For the plain Fibonacci matrix (deterministic) the exponent is just log((1+√5)/2), the golden ratio.

Why the hypotheses matter — and what breaks

Integrability is essential. If log⁺‖A‖ ∉ L¹ the limit (1/n) log‖Aⁿ‖ can be +∞ or fail to exist. Kingman's theorem itself requires the subadditive sequence to have an integrable positive part; drop it and even the top exponent is undefined. For the two-sided splitting you genuinely need log⁺‖A⁻¹‖ ∈ L¹ too — without it you keep the forward filtration but the backward directions can degenerate and the direct-sum splitting fails; you get only a flag.

Ergodicity is not essential — drop it and the theorem still holds, but λᵢ and mᵢ become T-invariant measurable functions of x rather than constants (constant on each ergodic component). Non-commutativity is the whole point: unlike scalars, ‖Aⁿ‖ ≠ ∏‖A‖ and eigenvalues of a product are not products of eigenvalues, which is exactly why singular values, not eigenvalues, drive the proof. The result connects to Furstenberg's positivity criterion, the Ledrappier–Young formula, and — via smooth cocycles A = DT — to Pesin theory.

Why it matters: Pesin theory, entropy, and beyond

Applied to the derivative cocycle A(x) = D_xT of a smooth map on a compact manifold (with μ a T-invariant probability), Oseledets produces Lyapunov exponents that measure exponential sensitivity to initial conditions — the rigorous meaning of chaos. This underpins Pesin theory: where exponents are nonzero, the unstable/stable subspaces integrate to invariant manifolds, and Pesin's entropy formula h_μ(T) = ∫ Σ λᵢ⁺ dμ (Ruelle's inequality h_μ ≤ ∫ Σ λᵢ⁺ becomes equality for SRB measures) ties exponents to Kolmogorov–Sinai entropy.

Beyond dynamics, Oseledets governs products of random matrices (Furstenberg–Kesten, transfer matrices in the theory of the Anderson model and 1-D random Schrödinger operators, where a positive exponent ⇒ Anderson localization), stability of random ODEs, and stochastic flows. Ruelle's 1982 extension to compact operators on Hilbert and Banach spaces reaches PDE dynamics and delay equations. It is the structural theorem that makes 'exponential growth rate' a well-defined, computable spectral invariant of non-autonomous linear dynamics.

How hypotheses and structure control the conclusion of the multiplicative ergodic theorem

Setting	Hypothesis on cocycle A	Conclusion you get
One-sided (non-invertible T or A ∈ Matₐ)	log⁺‖A‖ ∈ L¹	Flag / filtration V₁(x) ⊃ V₂(x) ⊃ ⋯ ; growth rate λᵢ on Vᵢ ∖ Vᵢ₊₁
Two-sided (invertible T, A ∈ GL(d))	log⁺‖A‖ and log⁺‖A⁻¹‖ ∈ L¹	Direct-sum splitting ℝᵈ = E₁(x) ⊕ ⋯ ⊕ E_k(x), forward AND backward rates λᵢ
Non-ergodic μ	Same integrability	Same result, but λᵢ(x) and multiplicities mᵢ(x) are T-invariant functions, not constants
Furstenberg–Kesten (1960, predecessor)	log⁺‖A‖ ∈ L¹	Only the top exponent λ₁ = lim (1/n) log‖Aⁿ(x)‖ exists a.e.
Hilbert / Banach space (Ruelle 1982)	A(x) compact / quasi-compact, log⁺‖A‖ ∈ L¹	Discrete exponents λ₁ ≥ λ₂ ≥ ⋯ ↓ κ (essential spectrum radius), finite-dim Eᵢ above κ