  Let $X$ be a proper geodesic metric space and let $G$ be a group of
isometries of $X$ which acts geometrically. Cordes constructed the Morse
boundary of $X$ which generalizes the contracting boundary for CAT(0) spaces
and the visual boundary for hyperbolic spaces. We characterize Morse elements
in $G$ by their fixed points on the Morse boundary $\partial_MX$. The dynamics
on the Morse boundary is very similar to that of a $\delta$-hyperbolic space.
In particular, we show that the action of $G$ on $\partial_MX$ is minimal if
$G$ is not virtually cyclic. We also get a uniform convergence result on the
Morse boundary which gives us a weak north-south dynamics for a Morse isometry.
This generalizes the work of Murray in the case of the contracting boundary of
a CAT(0) space.
