Covering Action on Conley Theory

TL;DR

This paper extends Conley index theory using covering spaces to analyze the dynamics of invariant sets, providing a generalized framework that includes connection matrices and Novikov differentials for non-isolated and infinite cyclic cases.

Contribution

It introduces a generalized Conley theory on covering spaces, enriching the analysis of invariant sets and unifying connection matrices with Novikov differentials.

Findings

Generalizes classical connection matrix theory.

Provides new tools for non-isolated invariant sets.

Links Novikov differential to $p$-connection matrices.

Abstract

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.