Linking Combinatorial and Classical Dynamics: Conley Index and Morse Decompositions

We prove that every combinatorial dynamical system in the sense of Forman, defined on a family of simplices of a simplicial complex, gives rise to a multivalued dynamical system  F on the geometric realization of the simplicial complex. Moreover, F may be chosen in such a way that the isolated invar...

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Bibliographic Details
Published inFoundations of computational mathematics Vol. 20; no. 5; pp. 967 - 1012
Main Authors Batko, Bogdan, Kaczynski, Tomasz, Mrozek, Marian, Wanner, Thomas
Format Journal Article
LanguageEnglish
Published New York Springer US 01.10.2020
Springer
Springer Nature B.V
Subjects
Applications of Mathematics
Combinatorial analysis
Computer Science
Decomposition
Dynamical systems
Economics
Fields (mathematics)
Linear and Multilinear Algebras
Math Applications in Computer Science
Mathematics
Mathematics and Statistics
Matrix Theory
Numerical Analysis
Secondary: 37B35
54H20
Conley theory
Conley–Morse graph
37E15
57M99
Isolated invariant set
57Q05
Morse decomposition
Discrete Morse theory
Simplicial complex
57Q15
Multivalued dynamical system
Combinatorial vector field
Primary: 37B30
Isolating block
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Summary:We prove that every combinatorial dynamical system in the sense of Forman, defined on a family of simplices of a simplicial complex, gives rise to a multivalued dynamical system  F on the geometric realization of the simplicial complex. Moreover, F may be chosen in such a way that the isolated invariant sets, Conley indices, Morse decompositions and Conley–Morse graphs of the combinatorial vector field give rise to isomorphic objects in the multivalued map case.
ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-020-09444-1