  The theorems of Kozlov and Sullivan characterize Gibbs measures as measures
with positive continuous specifications. More precisely, Kozlov showed that
every positive continuous specification on symbolic configurations of the
lattice is generated by a norm-summable interaction. Sullivan showed that every
shift-invariant positive continuous specification is generated by a
shift-invariant interaction satisfying the weaker condition of
variation-summability. These results were proven in the 1970s. An open question
since that time is whether Kozlov's theorem holds in the shift-invariant
setting, equivalently whether Sullivan's conclusion can be improved from
variation-summability to norm-summability. We show that the answer is no: there
exist shift-invariant positive continuous specifications that are not generated
by any shift-invariant norm-summable interaction. On the other hand, we give a
complete proof of an extension, suggested by Kozlov, of Kozlov's theorem to a
characterization of positive continuous specifications on configuration spaces
with arbitrary hard constraints. We also present an extended version of
Sullivan's theorem. Aside from simplifying some of the arguments in the
original proof, our new version of Sullivan's theorem applies in various
settings not covered by the original proof. In particular, it applies when the
support of the specification is the hard-core shift or the two-dimensional
$q$-coloring shift for $q\geq 6$.
