  We provide a definition of a $\prec$-asymptotic pair in a topological action
of a countable group $G$, where $\prec$ is an order on $G$ of type $\mathbb Z$.
We then prove that if $G$ is a countable amenable group and $(X,G)$ is a
topological $G$-action of positive entropy, then for every multiorder
$(\tilde{\mathcal O},\nu,G)$ and $\nu$-almost every order
$\prec\,\in\tilde{\mathcal O}$ there exists a $\prec$-asympotic pair in $X$.
This result is a generalization of the Blanchard-Host-Ruette Theorem for
classical topological dynamical systems (actions of~$\mathbb Z$). We also prove
that for every countable amenable group $G$, and every multiorder on $G$
arising from a tiling system, every topological $G$-action of entropy zero has
an extension which has no $\prec$-asymptotic pairs for any $\prec$ belonging to
this multiorder. Together, these two theorems give a characterization of
topological $G$-actions of entropy zero: $(X,G)$ has topological entropy zero
if and only if, for any multiorder $\tilde{\mathcal O}_{\boldsymbol{\mathsf
T}}$ on $G$ arising from a tiling system of entropy zero, there exists an
extension $(Y,G)$ of $(X,G)$, which has no $\prec$-asymptotic pairs for any
$\prec\,\in\tilde{\mathcal O}_{\boldsymbol{\mathsf T}}$, equivalently, there
exists a multiorder $(\tilde{\mathcal O},\nu,G)$ on $G$, such that for
$\nu$-almost any $\prec\,\in\tilde{\mathcal O}$, there are no
$\prec$-asymptotic pairs in $(Y,G)$.
