Infinite series in cohomology: attractors and Conley index

TL;DR

This paper investigates the cohomological Conley index of isolated invariant sets in dynamical systems, introducing a new cohomology series summation method to analyze attractor-repeller decompositions and compute fixed point indices.

Contribution

It provides a novel interpretation of the first cohomological Conley index, relates it to higher degrees, and develops a new cohomology series summation technique for complex unstable manifolds.

Findings

New computation methods for fixed point indices in 2D and 3D.

A classical problem is addressed with a new cohomology series approach.

The approach simplifies the analysis of unstable manifolds with complicated topology.

Abstract

In this paper we study the cohomological Conley index of arbitrary isolated invariant continua for continuous maps $f \colon U \subseteq \mathbb{R}^d \to \mathbb{R}^d$ by analyzing the topological structure of their unstable manifold. We provide a simple dynamical interpretation for the first cohomological Conley index, describing it completely, and relate it to the cohomological Conley index in higher degrees. A number of consequences are derived, including new computations of the fixed point indices of isolated invariant continua in dimensions 2 and 3. Our approach exploits certain attractor-repeller decomposition of the unstable manifold, reducing the study of the cohomological Conley index to the relation between the cohomology of an attractor and its basin of attraction. This is a classical problem that, in the present case, is particularly difficult because the dynamics is discrete and the topology of the unstable manifold can be very complicated. To address it we develop a new method that may be of independent interest and involves the summation of power series in cohomology: if $Z$ is a metric space and $K \subseteq Z$ is a compact, global attractor for a continuous map $g \colon Z \to Z$, we show how to interpret series of the form $\sum_{j \ge 0} a_j (g^*)^j$ as endomorphisms of the cohomology group of the pair $(Z,K)$.