Since seminal work of Bowen, it has been known that the specificationproperty implies various useful properties about a topological dynamicalsystem, among them uniqueness of the measure of maximal entropy (often referredto as intrinsic ergodicity). Weakenings of the specification property calledalmost weak specification and almost specification have been defined andprofitably applied in various works. However, it has been an open question whether either or both of theseproperties imply intrinsic ergodicity. We answer this question negatively byexhibiting examples of subshifts with multiple measures of maximal entropy withdisjoint support which have almost weak specification with any gap function$f(n) = O(\ln n)$ or almost specification with any mistake function $g(n) \geq4$. We also show some results in the opposite direction, showing that subshiftswith almost weak specification with gap function $f(n) = o(\ln n)$ or almostspecification with mistake function $g(n) = 1$ cannot have multiple measures ofmaximal entropy with disjoint support.
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