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Continuation of abstract Lyapunov graphs and the maximal number of Betti number variations


The study of a dynamical system is generally divided in two parts: understanding the recurrent behavior on the singularities and describing the gradient-like behavior elsewhere. Labelled Lyapunov graphs are combinatorial objects introduced by Franks in order to enlighten interactions between these two opposite features of the same dynamical system. In particular, the choice of the labels of Lyapunov graphs depends on the nature of the interaction we want to study. Bertolim, de Rezende e Mello presented a continuation result of an abstract Lyapunov graph to a Lyapunov graph of Morse type where the edges are labeled with Betti numbers of level sets and vertices are labeled with Conley homology indices. The proof of these results relied on network flow theory. However, in this same article it is shown that this continuation is not unique and a procedure to exhibit all possible continuations is established. Later Bertolim et al, with similar combinatorial techniques from network flow theory proved that graph continuation is a necessary condition for realization of the graph as a gradient flow on some manifold. In other words, this combinatorial study shed light on important topological consequences. One should highlight the importance of studying properties associated to abstract Lyapunov graphs which are registered on its labels since this has topological and dynamical significance, e.g., the realization as a gradient flow on a manifold. Moreover, once the graph is admissible it can be realized by a handle decomposition. A paper by Ogasa presents a topological invariant which consists in decomposing the manifold into the smallest number of handles and considering the variation of the sums of all Betti numbers after each gluing. In other word the variation of the Betti numbers of level sets. The Ogasa invariant is defined as the maximum among these sums at each level set. Apparently, in low dimensions, this is important in Quantum field theory in the study of Feynman diagrams. Since each edge on the graph is labelled with Betti numbers and since for each admissible graph a realization given by a handle decomposition can be associated, inspired on Ogasa's work we define an Ogasa number for abstract Lyapunov graphs. Our main goal is to enrich the labels of Lyapunov graphs with Ogasa invariants in order to obtain restrictions in their continuations. By associating to a Lyapunov graph some invariants which fix the maximal number of variation of the Betti numbers on the edges we will study the global topological consequences produced by this association. This study will be done combinatorially with network flow theory techniques, and thus advancing our knowledge of the dynamical and topological interaction. We will associate to a Lyapunov graph some invariants which fix the maximal number of variation of the Betti numbers on the edges and we will study the consequences produced by this association. (AU)

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(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
BERTOLIM, M. A.; DE REZENDE, K. A.; MANZOLI NETO, O.; VAGO, G. M. On the variations of the Betti numbers of regular levels of Morse flows. Topology and its Applications, v. 158, n. 6, p. 761-774, APR 1 2011. Web of Science Citations: 3.

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