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Oseledecs TheoremEdit

Oseledecs Theorem, often called the multiplicative ergodic theorem, is a foundational result in ergodic theory and dynamical systems that describes the long-term behavior of products of random matrices along orbits of a system equipped with an invariant measure. Under mild integrability hypotheses, it guarantees the almost-sure existence of Lyapunov exponents and a measurable invariant splitting of the ambient space into Oseledec subspaces, each associated with a concrete exponential growth rate. This provides a rigorous lens for understanding stability, chaos, and the geometry of growth in both deterministic and stochastic settings, with applications ranging from differential equations to statistical physics.

The core idea is that when one follows a trajectory of a system and multiplies a sequence of matrices along that trajectory, vectors experience exponential stretching or contraction. The rates of this stretching are captured by Lyapunov exponents, and the directions in which growth occurs form an invariant decomposition of the space. If the underlying system is ergodic, these growth rates become constants for almost every point in the space, making the theorem a powerful tool for predicting typical behavior across almost all starting conditions. The multiplicative ergodic theorem generalizes classical eigenvalue growth in linear systems to nonlinear, time-varying, and random contexts, tying local dynamics to global averages through measurable subspaces.

Theorem

Statement

Let (X, F, μ, T) be a probability space with T: X → X measure-preserving, and let A: X → GL(d, R) be a measurable map taking values in invertible d×d real matrices. Assume the integrability conditions - log^+ ||A(x)|| is μ-integrable, and - log^+ ||A(x)^{-1}|| is μ-integrable.

Define the cocycle A^n(x) = A(T^{n-1}x) · ... · A(Tx) · A(x) for n ≥ 1, and A^0(x) = I. Then there exist integers 1 ≤ p ≤ d and real numbers λ1 > λ2 > ... > λp and a measurable filtration of subspaces 0 ⊂ V1(x) ⊂ V2(x) ⊂ ... ⊂ Vp(x) = R^d such that for almost every x ∈ X, the subspaces Vi(x) have dimensions dim Vi(x) = i, and for every v ∈ Vi(x) \ Vi−1(x) (with V0(x) = {0}), lim_{n→∞} (1/n) log ||A^n(x) v|| = λi.

Moreover, the Oseledec splitting is invariant in the sense that A(x) Vi(x) ⊆ Vi(Tx), and the Lyapunov exponents are almost surely constant if μ is ergodic. When all exponents are distinct, this yields a direct sum decomposition into one-dimensional growth directions; in general, one works with the corresponding invariant subspaces Vi(x).

Invertible versus non-invertible cases

The classic formulation assumes A(x) ∈ GL(d, R) with the stated integrability conditions, which covers many invertible cocycles arising from differentiable dynamics and random matrix products. Extensions exist for certain non-invertible cocycles, but the precise statements and technical details differ. The core phenomenon—existence of almost-sure growth rates along invariant directions—persists in broad settings, though the structure of the splitting can be more delicate.

Ergodicity and regularity

If μ is ergodic, the Lyapunov exponents λi are nonrandom (i.e., constant μ-almost everywhere), which strengthens the predictive power of the theorem. The associated subspaces Vi(x) are typically only measurable in x, and not necessarily continuous, which influences both the theoretical interpretation and numerical computation in practice.

Foundations and related notions

Consequences and applications

  • Stability and hyperbolicity: The exponents separate directions that grow or contract at different exponential rates, underpinning hyperbolic behavior in many systems.
  • Invariant bundles: The Oseledec splitting provides a geometric lens on how phase space is organized under iteration, informing the structure of stable and unstable manifolds in smooth dynamics.
  • Entropy and dimension theory: Connections exist with entropy via formulas that relate growth rates to complexity measures (e.g., Pesin theory and related entropy formulas).
  • Random and time-dependent systems: The theorem applies to products of random matrices and to time-varying linearizations, informing models in physics, biology, and engineering.
  • Computation and estimation: Numerical methods seek to approximate Lyapunov exponents and subspaces, with practical algorithms inspired by the theoretical decomposition.

Generalizations and related results

  • Non-invertible cocycles: Extensions exist with adaptations to the growth rates and subspace structure.
  • Banach-space versions: There are multiplicative ergodic theorems in infinite-dimensional settings under suitable compactness or quasi-compactness conditions.
  • Random dynamical systems: The theorem underpins results in random and non-autonomous dynamics, where the base system introduces additional layers of complexity.

See also