Large deviations of Lyapunov exponents

Generic dynamical systems have 'typical' Lyapunov exponents, measuring the sensitivity to small perturbations of almost all trajectories. A generic system also has trajectories with exceptional values of the exponents, corresponding to unusually stable or chaotic situations. From a more mathematical point of view, large deviations of Lyapunov exponents characterize phase-space topological structures such as separatrices, homoclinic trajectories and degenerate tori. Numerically sampling such large deviations using the Lyapunov Weighted Dynamics allows one to pinpoint, for example, stable configurations in celestial mechanics or collections of interacting chaotic 'breathers' in nonlinear media. Furthermore, we show that this numerical method allows one to compute the topological pressure of extended dynamical systems, a central quantity in the thermodynamic of trajectories of Ruelle. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Lyapunov analysis: from dynamical systems theory to applications'. (paper)