the *(periodic) specification property of order $k$*. A dynamical system has (periodic) specification of order $k\in{\mathbb{N}}$ if for every ${\varepsilon}>0$ there is an $N$ such that every specification of rank $k$ is ${\varepsilon}$-traced by some (periodic) point. This weaker version of the (periodic) specification property may replace the stronger one in many proofs, but we do not know of any examples showing that these notions differ. We expect that even if they do, the examples demonstrating this would not be “natural”, that is, these potential examples would be systems defined for the sole purpose of proving that a specification property of finite order does not imply the specification property. Note that for shift spaces the periodic specification property, the specification property, and the specification property of order $k$, where $k\ge 2$ are equivalent (see Lemma \[lem:psp\] and Section \[sec:symbolic\]).

It is not hard to see that every map with the periodic specification property is onto, but this is not the case if the map has only specification.

Let $X={\left\{0,1\right\}}$ and $T\colon X\ni x\mapsto 0\in X$. Then $(X,T)$ has the specification property, but $T$ is not onto.

Every map on a one point space has the periodic specification property. Therefore we 