Asymptoticity, automorphism groups and strong orbit equivalence

Given any strong orbit equivalence class of minimal Cantor systems and any cardinal number that is finite, countable, or the continuum, we show that there exists a minimal subshift within the given class whose number of asymptotic components is exactly the given cardinal. For finite or countable ones, we explicitly construct such examples using $\mathcal{S}$-adic subshifts. We derived the uncountable case by showing that any topological dynamical system with countably many asymptotic components has zero topological entropy. We also construct systems with arbitrarily high subexponential word complexity with only one asymptotic class. We deduce that within any strong orbit equivalence class, there exists a subshift whose automorphism group is isomorphic to $\mathbb{Z}$.