On intrinsic ergodicity and weakenings of the specification property

Since seminal work of Bowen [2], it has been known that the specification property implies various useful properties about an expansive topological dynamical system, among them uniqueness of the measure of maximal entropy (often referred to as intrinsic ergodicity). Weakenings of the specification property have been defined and profitably applied in various works such as [6], [9], [11], [16] and [17].It has been an open question (see p. 798 of [4]) whether two of these properties, which we here call almost specification and non-uniform specification, imply intrinsic ergodicity for expansive topological systems. We answer this question negatively by exhibiting examples of subshifts with multiple measures of maximal entropy with disjoint support which have non-uniform specification with any gap function f(n)=O(ln⁡n)f(n)=O(ln⁡n) or almost specification with any mistake function g(n)≥4g(n)≥4. We also show some results in the opposite direction, showing that subshifts with non-uniform specification with gap function f(n)=o(ln⁡n)f(n)=o(ln⁡n) or almost specification with mistake function g(n)=1g(n)=1 cannot have multiple measures of maximal entropy with disjoint support.