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The Conley index for discrete dynamical systems and the mapping torus

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.

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Fig. 1

Notes

  1. 1.

    GAP–Groups, Algorithms, and Programming, Version 4.7.9 (2015), http://www.gap-system.org

  2. 2.

    Currently accessible at https://people.math.ethz.ch/~salamon/PREPRINTS/zeta.pdf

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Acknowledgements

The author would like to thank Marian Mrozek for numerous fruitful discussions during the development of this article and Dietmar Salamon for pointing to the unpublished preprint which was helpful for writing Sect. 7.

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Correspondence to Frank Weilandt.

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Weilandt, F. The Conley index for discrete dynamical systems and the mapping torus. J Appl. and Comput. Topology 3, 119–138 (2019). https://doi.org/10.1007/s41468-019-00027-w

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Keywords

  • Conley index
  • Shift equivalence
  • Mapping torus
  • Suspension flow

Mathematics Subject Classification

  • 37B30
  • 37B35