Collapsing of chaos in one dimensional maps

In their numerical investigation of the family of one dimensional maps fl(x)=1−2∣x∣l, where l>2, Diamond et al. [P. Diamond et al., Physica D 86 (1999) 559–571] have observed the surprising numerical phenomenon that a large fraction of initial conditions chosen at random eventually wind up at −1, a repelling fixed point. This is a numerical artifact because the continuous maps are chaotic and almost every (true) trajectory can be shown to be dense in [−1,1]. The goal of this paper is to extend and resolve this obvious contradiction. We model the numerical simulation with a randomly selected map. While they used 27 bit precision in computing fl, we prove for our model that this numerical artifact persists for an arbitrary high numerical prevision. The fraction of initial points eventually winding up at −1 remains bounded away from 0 for every numerical precision.