---
_id: '15064'
abstract:
- lang: eng
  text: We call a continuous self-map that reveals itself through a discrete set of
    point-value pairs a sampled dynamical system. Capturing the available information
    with chain maps on Delaunay complexes, we use persistent homology to quantify
    the evidence of recurrent behavior. We establish a sampling theorem to recover
    the eigenspaces of the endomorphism on homology induced by the self-map. Using
    a combinatorial gradient flow arising from the discrete Morse theory for Čech
    and Delaunay complexes, we construct a chain map to transform the problem from
    the natural but expensive Čech complexes to the computationally efficient Delaunay
    triangulations. The fast chain map algorithm has applications beyond dynamical
    systems.
acknowledgement: This research has been supported by the DFG Collaborative Research
  Center SFB/TRR 109 “Discretization in Geometry and Dynamics”, by Polish MNiSzW Grant
  No. 2621/7.PR/12/2013/2, by the Polish National Science Center under Maestro Grant
  No. 2014/14/A/ST1/00453 and Grant No. DEC-2013/09/N/ST6/02995. Open Access funding
  provided by Projekt DEAL.
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: U.
  full_name: Bauer, U.
  last_name: Bauer
- first_name: Herbert
  full_name: Edelsbrunner, Herbert
  id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
  last_name: Edelsbrunner
  orcid: 0000-0002-9823-6833
- first_name: Grzegorz
  full_name: Jablonski, Grzegorz
  id: 4483EF78-F248-11E8-B48F-1D18A9856A87
  last_name: Jablonski
  orcid: 0000-0002-3536-9866
- first_name: M.
  full_name: Mrozek, M.
  last_name: Mrozek
citation:
  ama: Bauer U, Edelsbrunner H, Jablonski G, Mrozek M. Čech-Delaunay gradient flow
    and homology inference for self-maps. <i>Journal of Applied and Computational
    Topology</i>. 2020;4(4):455-480. doi:<a href="https://doi.org/10.1007/s41468-020-00058-8">10.1007/s41468-020-00058-8</a>
  apa: Bauer, U., Edelsbrunner, H., Jablonski, G., &#38; Mrozek, M. (2020). Čech-Delaunay
    gradient flow and homology inference for self-maps. <i>Journal of Applied and
    Computational Topology</i>. Springer Nature. <a href="https://doi.org/10.1007/s41468-020-00058-8">https://doi.org/10.1007/s41468-020-00058-8</a>
  chicago: Bauer, U., Herbert Edelsbrunner, Grzegorz Jablonski, and M. Mrozek. “Čech-Delaunay
    Gradient Flow and Homology Inference for Self-Maps.” <i>Journal of Applied and
    Computational Topology</i>. Springer Nature, 2020. <a href="https://doi.org/10.1007/s41468-020-00058-8">https://doi.org/10.1007/s41468-020-00058-8</a>.
  ieee: U. Bauer, H. Edelsbrunner, G. Jablonski, and M. Mrozek, “Čech-Delaunay gradient
    flow and homology inference for self-maps,” <i>Journal of Applied and Computational
    Topology</i>, vol. 4, no. 4. Springer Nature, pp. 455–480, 2020.
  ista: Bauer U, Edelsbrunner H, Jablonski G, Mrozek M. 2020. Čech-Delaunay gradient
    flow and homology inference for self-maps. Journal of Applied and Computational
    Topology. 4(4), 455–480.
  mla: Bauer, U., et al. “Čech-Delaunay Gradient Flow and Homology Inference for Self-Maps.”
    <i>Journal of Applied and Computational Topology</i>, vol. 4, no. 4, Springer
    Nature, 2020, pp. 455–80, doi:<a href="https://doi.org/10.1007/s41468-020-00058-8">10.1007/s41468-020-00058-8</a>.
  short: U. Bauer, H. Edelsbrunner, G. Jablonski, M. Mrozek, Journal of Applied and
    Computational Topology 4 (2020) 455–480.
date_created: 2024-03-04T10:47:49Z
date_published: 2020-12-01T00:00:00Z
date_updated: 2024-03-04T10:54:04Z
day: '01'
ddc:
- '500'
department:
- _id: HeEd
doi: 10.1007/s41468-020-00058-8
file:
- access_level: open_access
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  creator: dernst
  date_created: 2024-03-04T10:52:42Z
  date_updated: 2024-03-04T10:52:42Z
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file_date_updated: 2024-03-04T10:52:42Z
has_accepted_license: '1'
intvolume: '         4'
issue: '4'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '12'
oa: 1
oa_version: Published Version
page: 455-480
publication: Journal of Applied and Computational Topology
publication_identifier:
  eissn:
  - 2367-1734
  issn:
  - 2367-1726
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Čech-Delaunay gradient flow and homology inference for self-maps
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 4
year: '2020'
...