Roundoff-induced attractors and reversibility in conservative two-dimensional maps

We numerically study two conservative two-dimensional maps, namely the baker map (whose Lyapunov exponent is known to be positive), and a typical one (exhibiting a vanishing Lyapunov exponent) chosen from the generalized shift family of maps introduced by C. Moore Phys. Rev. Lett. 64 (1990) 2354] in the context of undecidability. We calculate the time evolution of the entropy (). We exhibit the dramatic effect introduced by numerical precision. Indeed, in spite of being area-preserving maps, they present, well after the initially concentrated ensemble has spread virtually all over the phase space, unexpected pseudo-attractors (fixed-point like for the baker map, and more complex structures for the Moore map). These pseudo-attractors, and the apparent time (partial) reversibility they provoke, gradually disappear for increasingly large precision. In the case of the Moore map, they are related to zero Lebesgue-measure effects associated with the frontiers existing in the definition of the map. In addition to the above, and consistent with the results by V. Latora and M. Baranger Phys. Rev. Lett. 82 (1999) 520], we find that the rate of the far-from-equilibrium entropy production of baker map numerically coincides with the standard Kolmogorov-Sinai entropy of this strongly chaotic system.