  In this paper we introduce a notion of asymptotic stability of a probability
kernel, which we call dynamic uniqueness. We say that a kernel exhibits dynamic
uniqueness if all the stochastic chains starting from a fixed past coincide on
the future tail $\sigma$-algebra. We prove that the dynamic uniqueness is
generally stronger than the usual notion of uniqueness for $g$-measures. Our
main result shows that dynamic uniqueness is equivalent to the weak-$\ell^2$
summability condition on the kernel. This generalizes and strengthens the
Johansson-\"Oberg $\ell^2$ criterion for uniqueness of $g$-measures. Finally,
among other things, we prove that the weak-$\ell^2$ criterion implies
$\beta$-mixing of the unique $g$-measure compatible with a regular kernel
improving several results in the literature.
