The stability of saturated linear dynamical systems is undecidable

Abstract

We prove that several global properties (global convergence, global asymptotic stability, mortality, and nilpotence) of particular classes of discrete time dynamical systems are undecidable. Such results had been known only for point-to-point properties. We prove these proper- ties undecidable for saturated linear dynamical systems, and for con- tinuous piecewise a fine dynamical systems in dimension three. We also describe some consequences of our results on the possible dynamics of such systems.