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

Three billiards, whose border depends on a parameter ε, are considered; for ε = 0 they are integrable, while for ε > 0 they are known to be K-systems. The maximal Lyapunov exponent χ1 is numerically computed, and a power-law behavior χ1☆εβ 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 × 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 “universal” character. A heuristic “mean-field theory” is also proposed, based on the formal interpretation of the onset of stochasticity in our dynamical systems as a phase transition.