Deciding stability and mortality of piecewise affine dynamical systems

In this paper we study problems such as: given a discrete time dynamical system of the form x(t+1) = f(x(t)) where f:Rⁿ → Rⁿ is a piecewise affine function, decide whether all trajectories converge to 0. We show in our main theorem that this Attractivity Problem is undecidable as soon as n≥2. The same is true of two related problems: Stability (is the dynamical system globally asymptotically stable?) and Mortality (do all trajectories go through 0?). We then show that Attractivity and Stability become decidable in dimension 1 for continuous functions.