Adding machines and open dynamical systems

TL;DR

This paper investigates open dynamical systems on adding machines and solenoidal systems, showing that the set of indecisive trajectories is residual, highlighting complex behaviors in these simple systems.

Contribution

It introduces the concept of indecisive trajectories in open dynamical systems and proves their residuality in adding machine and solenoidal systems.

Findings

Indecisive trajectories form a residual set in adding machine systems.

Indecisive trajectories form a residual set in solenoidal systems.

The study extends understanding of complex behaviors in simple dynamical systems.

Abstract

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