theorem summarizes easy consequences of the (periodic) specification property.

1.  If $(X,T)$ has the (periodic) specification property then $(X,T^k)$ has the (periodic) specification property for every $k\geq 1$.

2.  If $(X,T)$, $(Y,S)$ have the (periodic) specification property then the product system $(X\times Y, T\times S)$ also has the (periodic) specification property.

3.  Every factor of a system with the (periodic) specification property has the (periodic) specification property.

4.  Every onto map with the specification property is topologically mixing.

The following fact is a simple consequence of the definition of expansiveness, but due to its importance we single it out as a separate lemma. It is proved implicitly by many authors, and an explicit statement and proof can be found as a part of Lemma 9 in [@KO12].

\[lem:psp\] If $(X,T)$ has the specification property and its natural extension is expansive, then $(X,T)$ has the periodic specification property.

Bowen [@Bowen71 Proposition 4.3] proved that any system with the periodic specification property[^2] on a nontrivial space has positive topological entropy with respect to any open cover of $X$ by two nondense open sets. In ergodic theory there is a class of $K$-systems [@Walters Definition 4.13], which contains measure preserving transformations whose measure-theoretic 