Integral representation of Lyapunov exponents

Abstract

We introduce a new operator-theoretic construction of Lyapunov growth for Markov-driven systems. The construction is based on an abstract variational principle for asymptotic growth rates arising from a subadditive process generated by an intertwining pair of Markov operators on a measurable bundle with compact fibers: for each invariant base measure, the fiberwise-maximal growth rate equals the supremum of the fiber integral over invariant lifts, and this supremum is attained on an ergodic lift. For random linear bundle morphisms driven by Markovian place-dependent noise, this yields Lyapunov exponents defined intrinsically from the initial law on the state-phase space. We prove that these exponents coincide with the classical path-space skew-product Lyapunov spectrum, showing that the pointwise exponents depend only on the current noise state and initial position, not on the full noise realization. We also obtain new asymptotic formulas for the sums of Lyapunov exponents as conditional annealed growth rates along individual directions. As further consequences of this variational principle, we recover and extend to singular linear bundle morphisms the classical projective variational formulas.