Linking Combinatorial and Classical Dynamics: Conley Index and Morse Decompositions
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Linking Combinatorial and Classical Dynamics: Conley Index and Morse Decompositions Bogdan Batko1 · Tomasz Kaczynski2 · Marian Mrozek1 · Thomas Wanner3 Received: 18 October 2017 / Revised: 15 August 2019 / Accepted: 4 November 2019 © The Author(s) 2020
Abstract 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. Keywords Combinatorial vector field · Multivalued dynamical system · Simplicial complex · Discrete Morse theory · Conley theory · Morse decomposition · Conley–Morse graph · Isolated invariant set · Isolating block Mathematics Subject Classification Primary: 37B30 · 54H20; Secondary: 37B35 · 37E15 · 57M99 · 57Q05 · 57Q15
1 Introduction In the years since Forman [14,15] introduced combinatorial vector fields on simplicial complexes, they have found numerous applications in such areas as visualization and mesh compression [21], graph braid groups [13], homology computation [17,25],
Communicated by Shmuel Weinberger. Research of B.B. and M.M. was partially supported by the Polish National Science Center under Maestro Grant No. 2014/14/A/ST1/00453. Research of T.K. was supported by a Discovery Grant from NSERC of Canada. T.W. was partially supported by NSF Grants DMS-1114923 and DMS-1407087, and by the Simons Foundation under Award 581334. All authors gratefully acknowledge the support of Hausdorff Research Institute for Mathematics in Bonn for providing an excellent environment to work together during the 2017 Special Hausdorff Program on Applied and Computational Algebraic Topology.
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Marian Mrozek [email protected]
Extended author information available on the last page of the article
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Foundations of Computational Mathematics
ˇ astronomy [38], the study of Cech and Delaunay complexes [6], and many others. One reason for this success has its roots in Forman’s original motivation. In his papers, he sought to transfer the rich dynamical theories due to Morse [26] and Conley [9] from the continuous setting of a continuum (connected compact metric space) to the finite, combinatorial setting of a simplicial complex. This has proved to be extremely useful for establishing finite, combinatorial results via ideas from dynamical systems. In particular, Forman’s theory yields an alternative when studying sampled dynamical systems. The classical approach consists in the numerical study of the dynamics of the differential equation constructed from the sample. The construction uses the data in the sample either to discover the natural laws governing the dynamics [37] in order to write the equations or to interpolate or approximate directly the unknown vector field in the differential equations [7].
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