Adding Machines and Open Dynamical Systems

Let \(f\colon \mathcal{M}\rightarrow\mathcal{M}\) be a continuous map defined on a compact metric space \(\mathcal{M}\). An open dynamical system introduces disjoint open balls centered at points in \(\mathcal{M}\), and considers the trajectories of points from \(\mathcal{M}\), and the balls that they visit first. As the balls in question are allowed to shrink, a point is considered indecisive if its trajectory changes infinitely many times the ball first visited. Here, we let \(\mathcal{M}\) be an adding machine, a simple system and a solenoidal system. In each case, we show that the set of points which generate indecisive trajectories is residual.