Collapsing of chaos in one dimensional maps

In their numerical investigation of the family of one dimensional maps f(l) (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 c ontinuous maps are chaotic and almost every (true) trajectory can be shown to he dense in [-1, 1]. The goal of this paper is to extend and resolve thi s obvious contradiction. We model the numerical simulation with a randomly selected map. While they used 27 bit precision in computing f(l), we prove for our model that this numerical artifact persists for an arbitrary high n umerical prevision. The fraction of initial points eventually winding up at -1 remains bounded away from 0 for every numerical precision. (C)2000 Else vier Science B.V. All rights reserved.