The Persistent Homology of a Self-Map

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...

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Bibliographic Details
Published inFoundations of computational mathematics Vol. 15; no. 5; pp. 1213 - 1244
Main Authors Edelsbrunner, Herbert, Jabłoński, Grzegorz, Mrozek, Marian
Format Journal Article
LanguageEnglish
Published New York Springer US 01.10.2015
Springer
Subjects
Analysis
Applications of Mathematics
Computer Science
Economics
Homology theory (Mathematics)
Linear and Multilinear Algebras
Mappings (Mathematics)
Math Applications in Computer Science
Mathematics
Mathematics and Statistics
Matrix Theory
Numerical Analysis
United States
55U99
Persistent homology
Algebraic algorithms
Stability
55N35
Category theory
37B99
18A99
Computational experiments
Discrete dynamical systems
Computational topology
Convergence
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Summary: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.
ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-014-9223-y