  We provide a definition of a $\prec$-asymptotic pair in a topological action
of a countable amenable group $G$, where $\prec$ is an order on $G$ of type
$\mathbb Z$. We then prove that if $(\tilde{\mathcal O},\nu,G)$ is a multiorder
on $G$, then for every topological $G$-action of positive entropy there exists
an $\prec$-asympotic pair for almost every order $\prec\,\in\tilde{\mathcal
O}$. This result is a generalization of the Blanchard-Host-Ruette Theorem for
classical topological dynamical systems (actions of $\mathbb Z$).
