The recurrence spectrum for dynamical systems beyond specification

Abstract

We introduce {\it (W')-specification} in terms of language decompositions of subshifts, and show that any recurrence set of a subshift with this property has full Hausdorff dimension. Our main result applies to a wide class of subshifts without specification, such as all $S$-gap shifts, some coded shifts, and the coding space of any transitive piecewise monotonic interval map with positive entropy. Further, for a wide class of piecewise expanding interval maps we show that any recurrence set has full Hausdorff dimension.