the Lyapunov exponent of a cocycle
/ lee-ah-POO-nof /
The Lyapunov exponent quantifies the exponential rate at which a dynamical system stretches (or contracts) along a typical orbit — the precise meaning of "sensitive dependence on initial conditions". When the stretching is described by a stationary sequence of linear maps, the top exponent measures how fast their product grows, and its existence is a clean consequence of Kingman's subadditive ergodic theorem.
The setting is a cocycle over a measure-preserving system (Omega, F, P, T): a measurable map A : Omega -> GL(d, R) (matrices), generating products A_n(omega) = A(T^(n-1) omega) ... A(T omega) A(omega). The Furstenberg-Kesten theorem says that if E[ log^+ ||A|| ] < infinity then (1/n) log ||A_n(omega)|| converges almost surely to an invariant limit; on an ergodic system this is a deterministic constant lambda_1, the top Lyapunov exponent. The proof is Kingman applied to the subadditive sequence g_n = log ||A_n||. More fully, the Oseledets multiplicative ergodic theorem refines this: almost every omega carries a filtration of subspaces along which the growth rates are lambda_1 >= lambda_2 >= ... >= lambda_d (the Lyapunov spectrum), a multiplicative analogue of the eigenvalue decomposition that holds for non-commuting random products.
Lyapunov exponents are the backbone of smooth dynamics and applications: lambda_1 > 0 is the rigorous fingerprint of chaos (nearby orbits separate exponentially); the sum of positive exponents controls entropy via the Pesin formula; products of random matrices model disordered media (Anderson localisation), and Lyapunov exponents of random walks on groups (Furstenberg's theory) drive rigidity results. Two honest cautions. First, the exponent is an almost-everywhere limit, not an orbit-by-orbit guarantee: exceptional orbits exist on a null set. Second, Kingman/Oseledets prove lambda_1 exists but say little about its value; Furstenberg's theorem gives a non-degeneracy criterion (under irreducibility and non-compactness, lambda_1 > 0 strictly), but computing exponents exactly is generically intractable.
A simple 1D analogue: if A(omega) is multiplication by a positive scalar a(omega) with E[ log a ] finite, then (1/n) log |A_n| = (1/n) sum log a(T^k omega) -> E[ log a ] by Birkhoff — the Lyapunov exponent is just the mean of log a. The cocycle/matrix case is the non-commutative generalisation where the sum becomes a matrix product and Birkhoff becomes Kingman.
Scalar case: exponent = mean of log (Birkhoff). Matrix case: exponent = subadditive limit of log-norm (Kingman / Furstenberg-Kesten).
Matrix multiplication does not commute, so lambda_1 is not the average of the log-norms of individual matrices; the order of the product matters and the exponent is generally strictly less than that naive average.