The Conley index for discrete dynamical systems and the mapping torus

TL;DR

This paper introduces a new topological invariant for discrete dynamical systems, the homotopy type of the mapping torus of the index map, offering a novel approach for comparing Conley indices both theoretically and numerically.

Contribution

It proposes using the homotopy type of the mapping torus as an alternative invariant for Conley indices in discrete systems, expanding the tools for analysis and computation.

Findings

The homotopy type of the mapping torus can distinguish Conley indices.

Comparison with shift equivalence shows new insights.

Sketches a method for rigorous numerical construction.

Abstract

The Conley index for flows is a topological invariant describing the behavior around an isolated invariant set $S$. It is defined as the homotopy type of a quotient space $N/L$, where $(N,L)$ is an index pair for $S$. In the case of a discrete dynamical system, i.e., a continuous self-map $f\colon X\to X$, the definition is similar. But one needs to consider the index map $f_{(N,L)}\colon N/L\to N/L$ induced by $f$. The Conley index in this situation is defined as the homotopy class $[f_{(N,L)}]$ modulo shift equivalence. The shift equivalence relation is rarely used outside this context and not well understood in general. For practical purposes like numerical computations, one needs to use weaker algebraic invariants for distinguishing Conley indices, usually homology. Here we consider a topological invariant: the homotopy type of the mapping torus of the index map $f_{(N,L)}$. Using a homotopy type offers new ways for comparing Conley indices -- theoretically and numerically. We present some basic properties and examples, compare it to the definition via shift equivalence and sketch an idea for its construction using rigorous numerics.