Power-law behavior of Lyapunov exponents in some conservative dynamical systems

Three billiards, whose border depends on a parameter \u3b5, are considered; for \u3b5 = 0 they are integrable, while for \u3b5 > 0 they are known to be K-systems. The maximal Lyapunov exponent \u3c71 is numerically computed, and a power-law behavior \u3c71&06\u3b5\u3b2 is found, with View the MathML source for all billiards. A related abstract problem is then considered, precisely the case of an infinite product of conservative 2 7 2 random matrices, which are perturbations of commuting parabolic ones. Analogous computations give here two power-laws, with View the MathML source or View the MathML source, depending on the probability law used to construct random matrices. All these phenomena seem to have an \u201cuniversal\u201d character. A heuristic \u201cmean-field theory\u201d is also proposed, based on the formal interpretation of the onset of stochasticity in our dynamical systems as a phase transition