Shadowing With Small Diameter Sets Of Bounded Cardinality

We examine dynamical systems with the property that pseudo-orbits can be traced by small diameter sets with bounded cardinality. In particular, we show that mixing sofic subshifts and surjective dynamical systems with the specification property have this property, and in these systems, it is sufficient to consider small sets of cardinality no more than two. We also prove that a more general class of subshifts, the subshifts of quasi-finite type, exhibit this property.