Definition
Lyapunov exponent of a linear cocycle
An asymptotic exponential growth rate of a vector under iterated linear dynamics.
For a linear cocycle and a nonzero vector , the Lyapunov exponent, when the limit exists, is
It measures exponential growth or contraction along the orbit. Equivalent fixed norms in finite dimension give the same limit.
Relation to Koopman frequencies
A positive Lyapunov exponent concerns growth of vectors, for example infinitesimal perturbations of a trajectory. A Koopman eigenvalue of a probability-preserving invertible system has modulus one and describes oscillation of an observable. These quantities answer different questions and can coexist in a mixing system such as the cat map.