  Let $\mathcal{A}$ be an additive $k-$category and $\mathbf{C}_{\equiv
m}(\mathcal{A})$ be the category of $m-$periodic objects. For any integer
$m>1$, we study conditions under which the compression functor ${\mathcal F}_m
:\mathbf{C}^{b}(\mathcal{A}) \rightarrow \mathbf{C}_{\equiv m}(\mathcal{A})$
preserves or reflects irreducible morphisms. Moreover, we find sufficient
conditions for the functor ${\mathcal F}_m $ to be a Galois $G$-covering in the
sense of \cite{BL}. If in addition $\mathcal{A}$ is a dualizing category and
$\mbox{mod}\, \mathcal{A}$ has finite global dimension then $\mathbf{C}_{\equiv
m}(\mathcal{A})$ has almost split sequences. In particular, for a finite
dimensional algebra $A$ with finite strong global dimension we determine how to
build the Auslander-Reiten quiver of the category $\mathbf{C}_{\equiv
m}(\mbox{proj}\, A)$. Furthermore, we study the behavior of sectional paths in
$\mathbf{C}_{\equiv m}(\mbox{proj}\, A)$, whenever $A$ is any finite
dimensional $k-$algebra over a field $k$.
