Characterization of the stability of chains associated with $g$-measures

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.