{"abstract":[{"text":"Considering a continuous self-map and the induced endomorphism on homology, we study the eigenvalues and eigenspaces of the latter. Taking a filtration of representations, we define the persistence of the eigenspaces, effectively introducing a hierarchical organization of the map. The algorithm that computes this information for a finite sample is proved to be stable, and to give the correct answer for a sufficiently dense sample. Results computed with an implementation of the algorithm provide evidence of its practical utility.\r\n","lang":"eng"}],"has_accepted_license":"1","language":[{"iso":"eng"}],"pubrep_id":"486","isi":1,"scopus_import":"1","department":[{"_id":"HeEd"}],"title":"The persistent homology of a self-map","ec_funded":1,"type":"journal_article","month":"10","author":[{"full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833"},{"last_name":"Jablonski","first_name":"Grzegorz","orcid":"0000-0002-3536-9866","full_name":"Jablonski, Grzegorz","id":"4483EF78-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Marian","last_name":"Mrozek","full_name":"Mrozek, Marian"}],"oa_version":"Published Version","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"doi":"10.1007/s10208-014-9223-y","file":[{"date_created":"2018-12-12T10:08:10Z","relation":"main_file","content_type":"application/pdf","date_updated":"2020-07-14T12:45:26Z","file_id":"4670","checksum":"3566f3a8b0c1bc550e62914a88c584ff","file_size":1317546,"access_level":"open_access","file_name":"IST-2016-486-v1+1_s10208-014-9223-y.pdf","creator":"system"}],"license":"https://creativecommons.org/licenses/by/4.0/","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","status":"public","date_published":"2015-10-01T00:00:00Z","acknowledgement":"This research is partially supported by the Toposys project FP7-ICT-318493-STREP, by ESF under the ACAT Research Network Programme, by the Russian Government under mega project 11.G34.31.0053, and by the Polish National Science Center under Grant No. N201 419639.","project":[{"name":"Topological Complex Systems","call_identifier":"FP7","_id":"255D761E-B435-11E9-9278-68D0E5697425","grant_number":"318493"}],"_id":"2035","publication_status":"published","publist_id":"5022","oa":1,"date_created":"2018-12-11T11:55:20Z","page":"1213 - 1244","file_date_updated":"2020-07-14T12:45:26Z","date_updated":"2025-09-23T14:08:54Z","year":"2015","citation":{"apa":"Edelsbrunner, H., Jablonski, G., &#38; Mrozek, M. (2015). The persistent homology of a self-map. <i>Foundations of Computational Mathematics</i>. Springer. <a href=\"https://doi.org/10.1007/s10208-014-9223-y\">https://doi.org/10.1007/s10208-014-9223-y</a>","short":"H. Edelsbrunner, G. Jablonski, M. Mrozek, Foundations of Computational Mathematics 15 (2015) 1213–1244.","chicago":"Edelsbrunner, Herbert, Grzegorz Jablonski, and Marian Mrozek. “The Persistent Homology of a Self-Map.” <i>Foundations of Computational Mathematics</i>. Springer, 2015. <a href=\"https://doi.org/10.1007/s10208-014-9223-y\">https://doi.org/10.1007/s10208-014-9223-y</a>.","ista":"Edelsbrunner H, Jablonski G, Mrozek M. 2015. The persistent homology of a self-map. Foundations of Computational Mathematics. 15(5), 1213–1244.","mla":"Edelsbrunner, Herbert, et al. “The Persistent Homology of a Self-Map.” <i>Foundations of Computational Mathematics</i>, vol. 15, no. 5, Springer, 2015, pp. 1213–44, doi:<a href=\"https://doi.org/10.1007/s10208-014-9223-y\">10.1007/s10208-014-9223-y</a>.","ama":"Edelsbrunner H, Jablonski G, Mrozek M. The persistent homology of a self-map. <i>Foundations of Computational Mathematics</i>. 2015;15(5):1213-1244. doi:<a href=\"https://doi.org/10.1007/s10208-014-9223-y\">10.1007/s10208-014-9223-y</a>","ieee":"H. Edelsbrunner, G. Jablonski, and M. Mrozek, “The persistent homology of a self-map,” <i>Foundations of Computational Mathematics</i>, vol. 15, no. 5. Springer, pp. 1213–1244, 2015."},"article_processing_charge":"No","day":"01","publisher":"Springer","volume":15,"quality_controlled":"1","external_id":{"isi":["000360862900004"]},"publication":"Foundations of Computational Mathematics","intvolume":"        15","ddc":["000"],"issue":"5"}