  We prove that uniqueness of the stationary chain, or equivalently, of the
$g$-measure, compatible with an attractive regular probability kernel is
equivalent to either one of the following two assertions for this chain: (1) it
is a finitary coding of an i.i.d. process with countable alphabet, (2) the
concentration of measure holds at exponential rate. We show in particular that
if a stationary chain is uniquely defined by a kernel that is continuous and
attractive, then this chain can be sampled using a coupling-from-the-past
algorithm. For the original Bramson-Kalikow model we further prove that there
exists a unique compatible chain if and only if the chain is a finitary coding
of a finite alphabet i.i.d. process. Finally, we obtain some partial results on
conditions for phase transition for general chains of infinite order.
