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The Conley index for discrete dynamical systems and the mapping torus

The Conley index for discrete dynamical systems and the mapping torus The classical Conley index for flows is defined as a certain homotopy type. In the case of a discrete dynamical system, one usually considers the shift equivalence class of the so-called index map. This equivalence relation is rarely used in other contexts and not well understood in general. Here we propose using a topological invariant of the shift equivalence definition: The homotopy type of the mapping torus of the index map. 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Applied and Computational Topology Springer Journals

The Conley index for discrete dynamical systems and the mapping torus

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Publisher
Springer Journals
Copyright
Copyright © 2019 by Springer Nature Switzerland AG
Subject
Mathematics; Algebraic Topology; Computational Science and Engineering; Mathematical and Computational Biology
ISSN
2367-1726
eISSN
2367-1734
DOI
10.1007/s41468-019-00027-w
Publisher site
See Article on Publisher Site

Abstract

The classical Conley index for flows is defined as a certain homotopy type. In the case of a discrete dynamical system, one usually considers the shift equivalence class of the so-called index map. This equivalence relation is rarely used in other contexts and not well understood in general. Here we propose using a topological invariant of the shift equivalence definition: The homotopy type of the mapping torus of the index map. 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.

Journal

Journal of Applied and Computational Topology – Springer Journals

Published: Jun 18, 2019

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