  In [2], Bowen showed that for an expansive system (X, T) with specification
and a potential with the Bowen property, the equilibrium state is unique and
fully supported. We generalize that result by showing that the same conclusion
holds for non-uniform versions of Bowen's hypotheses in which constant
parameters are replaced by any increasing unbounded functions f(n) and g(n)
with sublogarithmic growth (in n). We prove results for two weakenings of
specification; the first is non-uniform specification, based on a definition of
Marcus in ([14]), and the second is a significantly weaker property which we
call non-uniform transitivity. We prove uniqueness of the equilibrium state in
the former case under the assumption that liminf (f(n) + g(n))/ln n = 0, and in
the latter case when lim (f(n) + g(n))/ln n = 0. In the former case, we also
prove that the unique equilibrium state has the K-property. It is known that
when f(n)/ln n or g(n)/ln n is bounded from below, equilibrium states may not
be unique, and so this work shows that logarithmic growth is in fact the
optimal transition point below which uniqueness is guaranteed. Finally, we
present some examples for which our results yield the first known proof of
uniqueness of equilibrium state.
