Linking Combinatorial and Classical Dynamics: Conley Index and Morse Decompositions

Foundations of Computational Mathematics https://doi.org/10.1007/s10208-020-09444-1 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. B Marian Mrozek Extended author information available on the last page of the article 123 Foundations of Computational Mathematics astronomy [38], the study of Čech 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]. In the emerging alternative, one can eliminate differential equations and study directly the combinatorial dynamics defined by the sample [14,15,20,33,34]. The two approaches are essentially distinct. On the one hand, dynamical systems defined by differential equations on a differentiable manifold arise in a wide variety of applications and show an extreme wealth of observable dynamical behavior, at the expense of fairly involved mathematical techniques which are needed for their precise description. On the other hand, the discrete simplicial complex setting makes the study of many phenomena simple, due to the availability of fast combinatorial algorithms. This leads to the natural question of which approach should be chosen when for a given problem. In order to answer this question, it may be helpful to go beyond the exchange of abstract underlying ideas present in much of the existing work and look for the precise relation between the two theories. In our previous paper [19], we took this path and studied the formal ties of multivalued dynamics in the combinatorial and continuum settings. The choice of multivalued dynamics is natural because the combinatorial vector fields generate multivalued dynamics in a natural way. Moreover, in the finite setting such dynamical phenomena as homoclinic or heteroclinic connections are not possible in single-valued dynamics. The choice of multivalued dynamics on continua is not a restriction. This is a broadly studied and well-understood theory. The theory originated in the middle of the twentieth century from the study of contingent equations and differential inclusions [3,36,42] and control theory [35]. At the end of the twentieth century, it was successfully applied to computer-assisted proofs in dynamics [24,32]. In particular, the Conley theory for multivalued dynamics was studied by several authors [4,5,11,12,18,28,40]. In [19] we proved that for any combinatorial vector field on the collection of simplices of a simplicial complex, one can construct an acyclic-valued and upper semicontinuous map on the underlying geometric realization whose dynamics on the level of invariant sets exhibits the same complexity. More precisely, by introducing the notion of isolated invariant sets in the discrete setting, we could show that every isolated invariant set of the combinatorial vector field gives rise to a corresponding isolated invariant set in the classical multivalued setting. We also presented a link between the combinatorial and the classical multivalued dynamics on the level of individual dynamical trajectories. 123 Foundations of Computational Mathematics Fig. 1 Sample discrete vector field. This figure shows a simplicial complex X which is a graph on six vertices with seven edges. Critical cells are indicated by red dots; vectors of the vector field are shown as red arrows (Color figure online) In the present paper, we complete the program started in [19] by showing that the above-described correspondence extends to Conley indices of the corresponding isolated invariant set as well as Morse decompositions and Conley–Morse graphs [2,8], a global descriptor of dynamics capturing its gradient structure. The organization of the paper is as follows: In Sect. 2 we present the main result of the paper and illustrate it with some examples. In Sect. 3 we recall the basics of the Conley theory for multivalued dynamics. In Sect. 4 we recall from [19] the construction of a multivalued self-map F : X  X associated with a combinatorial vector field V on a simplicial complex X with the geometric realization X := |X |. In Sect. 5 we use this construction to outline the proof of the main result of the paper in a series of auxiliary theorems. The remaining sections are devoted to the proofs of these theorems. 2 Main Result Let X denote the family of simplices of a fi (...truncated)