  It is well-known that for expansive maps and continuous potential functions,
the specification property (for the map) and the Bowen property (for the
potential) together imply the existence of a unique equilibrium state. We
consider symbolic spaces that may not have specification, and potentials that
may not have the Bowen property, and give conditions under which uniqueness of
the equilibrium state can still be deduced. Our approach is to ask that the
collection of cylinders which are obstructions to the specification property or
the Bowen property is small in an appropriate quantitative sense. This allows
us to construct an ergodic equilibrium state with a weak Gibbs property, which
we then use to prove uniqueness. We do not use inducing schemes or the
Perron--Frobenius operator, and we strengthen some previous results obtained
using these approaches. In particular, we consider $\beta$-shifts and show that
the class of potential functions with unique equilibrium states strictly
contains the set of potentials with the Bowen property. We give applications to
piecewise monotonic interval maps, including the family of geometric potentials
for examples which have both indifferent fixed points and a non-Markov
structure.
