the remaining set ${\mathcal D}_{\mathsf{cofinite}}\setminus{\mathcal D}_{\mathsf{free}}$ it is neither infinitely conservative nor dissipative.

The group ${\mathbb Z}$ does not contain non-trivial finite subgroups, so that for its actions ${\mathcal D}_{\mathsf{cofinite}}={\mathcal D}_{\mathsf{free}}$, and therefore infinite conservativity is equivalent to plain conservativity. The conservative part of the action in this case is the union of the continual part ${\mathcal C}$ and the set of all the periodic points ${\mathcal D}\setminus{\mathcal D}_{\mathsf{cofinite}}$.

When dealing with the actions of the group ${\mathbb Z}$ one sometimes defines the notion of recurrence by looking only at the “positive semi-orbits” ${\mathbb Z}_+x$. All measurable subsets of ${\mathcal C}$ are recurrent in this sense as well. Indeed, for a subset $A\subset{\mathcal C}$ let $A_0=\{x\in A: {\mathbb Z}_+x\cap A \;\text{is
finite}\}$. Then $x\mapsto zx$, where $z$ is the maximal element of ${\mathbb Z}$ with $zx\in A$, is a measurable “selection map” in the sense of [Lemma \[lem:choice\]]{}, so that $A_0\subset{\mathcal D}$, whence $A_0$ is a null set.

If one defines *strict recurrence* and *strict conservativity* by requiring that the orbit $Gx$ returns to the set $A$ at an orbit point *different* from the starting point $x$, then the strictly conservative part of the action coincides just with the continual 