Oseledets theorem

The Oseledets theorem, also known as the multiplicative ergodic theorem, is the foundational result that guarantees the existence of Lyapunov exponents for dynamical systems preserving an invariant measure. Proved by Valery Oseledets in 1965, it establishes that for almost every point in phase space, the tangent space decomposes into invariant subspaces — the Oseledets splitting — each expanding or contracting at a rate given by a Lyapunov exponent.

The theorem transforms Lyapunov exponents from numerical observations into rigorous geometric invariants. Without it, the entire edifice of Pesin theory and the Ledrappier-Young formula would rest on heuristic ground. The proof relies on the subadditive ergodic theorem and yields a decomposition that is measurable but not necessarily continuous — a subtlety that has profound consequences for the global geometry of non-uniformly hyperbolic systems.

The Oseledets theorem has been extended to random dynamical systems, cocycles over measure-preserving transformations, and infinite-dimensional settings. Its generalizations are central to the ergodic theory of partially hyperbolic systems and to the spectral theory of transfer operators in the thermodynamic formalism.

The Oseledets theorem is not a technical prerequisite for chaos theory; it is the declaration that chaos has a geometry, and that this geometry can be measured.