Intrinsic ergodicity beyond specification: beta-shifts, S-gap shifts, and their factors

We give sufficient conditions for a shift space $(\Sigma,\sigma)$ to be intrinsically ergodic, along with sufficient conditions for every subshift factor of $\Sigma$ to be intrinsically ergodic. As an application, we show that every subshift factor of a $\beta$-shift is intrinsically ergodic, which answers an open question included in Mike Boyle's article "Open problems in symbolic dynamics". We obtain the same result for $S$-gap shifts, and describe an application of our conditions to more general coded systems. One novelty of our approach is the introduction of a new version of the specification property that is well adapted to the study of symbolic spaces with a non-uniform structure.