Lyapunov exponents for random maps (Q2082013)

The Lyapunov exponents exist almost everywhere with respect to any invariant measure, according to the celebrated theorem of \textit{V. I. Oseledets} [Trans. Mosc. Math. Soc. 19, 197--231 (1968; Zbl 0236.93034); translation from Tr. Mosk. Mat. Obshch. 19, 179--210 (1968)]. However, recent progress in the field has shown that there is an abundant class of systems for which the Lyapunov irregular set (i.e., the set where the Lyapunov exponent fails to exist) has positive Lebesgue measure. The present paper shows that the situation can be rectified by introducing a small amount of noise to the system. Specifically, for any suitably chosen random perturbation of the system, the Lyapunov irregular set has zero Lebesgue measure for almost every realization of the system. Furthermore, the authors prove that the considered random system admits finitely many physical absolutely continuous ergodic measures (in [\textit{V. Araújo}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 17, No. 3, 307--369 (2000; Zbl 0974.37036)] a similar proof strategy was used). The paper also contains numerical computations of the Lyapunov exponents for surface flows that are known to lack Lyapunov exponents. Finally, the authors provide an example where Lyapunov exponents exist under impulsive types of perturbations.