Adding machines and open dynamical systems

Abstract

Let $f:\mathcal{M}\to \mathcal{M}$ 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. Keywords: Open dynamical system, Adding machine, Solenoidal system, Baire category