Ergodic Properties of High-dimensional Symplectic Maps

We report extensive numerical studies on the long-time behavior of a high-dimensional system of coupled symplectic maps as a function of their number N and of the nearest-neighbor coupling strength epsilon. The system, at a fixed value of epsilon, displays regular motion only in a small fraction of the phase space, whose volume vanishes exponentially with N. Regarding the chaotic motion, we find a scaling behavior of the mean-square fluctuation sigma of the maximal Lyapunov exponent about its average value over initial conditions: sigma congruent-to (1/N)alpha where alpha = O(unroofed-radical epsilonBAR). Nevertheless, also for large systems, one observes a very weak Arnold diffusion, and different trajectories, with a high value of the Lyapunov exponents, maintain some of their own features for a very long time. Finally, we study the localization properties of the tangent vector. For chaotic trajectories, at small values of epsilon, an initially small perturbation increases only in a few directions; due to the translational invariance of the system, this behavior may be seen as a failure of ergodicity and also as a confirmation of the relevance of the Nekhoroshev scenario in high-dimensional systems.