  In this paper, we apply Conley index theory in a covering space of an
invariant set $S$, possibly not isolated, in order to describe the dynamics in
$S$. More specifically, we consider the action of the covering translation
group in order to define a topological separation of $S$ which distinguish the
connections between the Morse sets within a Morse decomposition of $S$. The
theory developed herein generalizes the classical connection matrix theory,
since one obtains enriched information on the connection maps for non isolated
invariant sets, as well as, for isolated invariant sets. Moreover, in the case
of the infinite cyclic covering induced by a circle-valued Morse function, one
proves that Novikov differential of $f$ is a particular case of the
$p$-connection matrix defined herein.

Dynamical Systems;Algebraic Topology