Collapsing of chaos in one dimensional maps (Q1961654)

Considering the family of one-dimensional maps \(f_l(x)=1-2|x|^l\), \(l>2\), the authors investigate the numerical artifact when a large fraction of numerically computed orbits wind up at a repelling fixed point [\textit{P. Diamond, M. Suzuki, P. Kloeden}, and \textit{P. Pokrovskii}, Comput. Math. Appl. 31, No. 11, 83-95 (1996; Zbl 0854.34043)]. This is the case in which numerical simulations yield incorrect results since the maps are chaotic and almost every trajectory is dense in \([-1,1]\). The authors prove that this artifact persists for an arbitrary high numerical precision, i.e. fraction of initial points eventually winding up at the repelling point remains bounded away from zero. In particular, the lower bound for this fraction is given as \(\delta^{1/2}\) for \(l=2+\delta\) and \(1\) for \(l\to\infty\). The average collapsing time is also estimated.