Mathematical theory of Lyapunov exponents

This paper reviews some basic mathematical results on Lyapunov exponents, one of the most fundamental concepts in dynamical systems. The first few sections contain some very general results in nonuniform hyperbolic theory. We consider (f, mu), where f is an arbitrary dynamical system and mu is an arbitrary invariant measure, and discuss relations between Lyapunov exponents and several dynamical quantities of interest, including entropy, fractal dimension and rates of escape. The second half of this review focuses on observable chaos, characterized by positive Lyapunov exponents on positive Lebesgue measure sets. Much attention is given to SRB measures, a very special kind of invariant measures that offer away to understand observable chaos in dissipative systems. Paradoxical as it may seem, given a concrete system, it is generally impossible to determine with mathematical certainty if it has observable chaos unless strong geometric conditions are satisfied; case studies will be discussed. The final section is on noisy or stochastically perturbed systems, for which we present a dynamical picture simpler than that for purely deterministic systems. In this short review, we have elected to limit ourselves to finite-dimensional systems and to discrete time. The phase space, which is assumed to be R-d or a Riemannian manifold, is denoted by M throughout. The Lebesgue or the Riemannian measure on M is denoted by m, and the dynamics are generated by iterating a self-map of M, written f : M (sic). For flows, the reviewed results are applicable to time-t maps and Poincare return maps to cross-sections.