  Given a group $G$ acting geometrically on a systolic complex $X$ and a
hyperbolic isometry $h \in G$, we study the associated action of $h$ on the
systolic boundary $\partial X$. We show that $h$ has a canonical pair of fixed
points on the boundary and that it acts trivially on the boundary if and only
if it is virtually central. The key tool that we use to study the action of $h$
on $\partial X$ is the notion of a $K$-displacement set of $h$, which
generalises the classical minimal displacement set of $h$. We also prove that
systolic complexes equipped with a geometric action of a group are almost
extendable.
