Lyapunov graph continuation

Abstract

In this paper the Poincaré-Hopf inequalities are shown to be necessary and sufficient conditions for an abstract Lyapunov graph L to be continued to an abstract Lyapunov graph of Morse type. The Lyapunov graphs considered may represent smooth flows on closed orientable n-manifolds, n ≥ 2. The continuation which is presented by means of a constructive algorithm, is shown to be unique in dimensions two and three. In all other dimensions, the exact number of possible continuations of L are presented. Downloaded: 05 Apr 2015 IP address: 150.131.192.151 2 M. A. Bertolim et al Lyapunov graphs were first introduced by Franks [6]and have proven to be an excellent bookkeeping device of dynamical and topological information of the flow and the phase space. A Lyapunov graph is a finite, connected, oriented graph with no oriented cycles and with labeled vertices and edges.A Lyapunov function f : M → R associated to a flow, determines a Lyapunov graph by the following equivalence relation on M: x ∼ f y if and only if x and y belong to the same connected component of a level set of f . Therefore, M/ ∼ f is a Lyapunov graph. It is possible to choose f so that each critical level contains a component R of R. A point on M/ ∼ f is a vertex point if under the equivalence relation it corresponds to a level set containing a component R of R. All other points are edge points. Each edge represents a codimension one submanifold Q of M times an open bounded interval I , Q × I .In order to retain some topological information of Q × I , the edges will be labeled with the Betti numbers of Q. All homology groups will be computed with Z 2 coefficients.An abstract Lyapunov graph is an oriented graph with no oriented cycles such that each vertex v is labeled with a list of non-negative integersWhenever h j (v) = 0, it will be omitted from the list. Also, the labels on each edge {β 0 = 1, β 1 , . . . , β n−2 , β n−1 = 1} must be a collection of non-negative integers satisfying Poincaré duality and if n − 1 is even then β (n−1)/2 is even.An abstract Lyapunov graph of Morse type is defined later, but roughly speaking, it is an abstract Lyapunov graph with all vertices labeled with non-degenerate singularities, i.e. {h j (v) = 1} for some j .Next, the notion of vertex explosion is established in order to define continuation of abstract Lyapunov graphs. Let v be a vertex on an abstract Lyapunov graph labeled with {h 0 (v), h 1 (v), . . . , h n (v)}. A vertex v can be exploded if v can be removed and replaced by an abstract Lyapunov graph I of Morse type with k vertices. The graph I must respect the orientations and labels of the incoming and outgoing edges of v. In other words, the new graph obtained must be oriented and with no oriented cycles. The incoming (outgoing) edges of v, must be incoming (outgoing) edges on vertices of I and all labels on the edges must respect the restrictions of the Morse type vertices. Moreover,An abstract Lyapunov graph admits a continuation to an abstract Lyapunov graph of Morse type if each vertex can be exploded.The...

Poincaré-Hopf inequalities

“…We also show in [1] that (2.6) and (2.7) and consequently the h cd -system above, constitute a network-flow problem with possible additional constraints. The nature of the network involved allows a complete characterization of all possible solutions of the h cd -system by means of one particular (complementary) solution of the system (h A natural problem to consider is the calculation of all possible Betti number vectors that satisfy the Morse inequalities for a given pre-assigned index data (h 0 , .…”

“…This article is divided in the following sections. Section 2 will summarize the Poincaré-Hopf inequalities for isolating blocks obtained in [1]. In Section 3 we prove Theorem 1.1 in several stages: Subsection 3.1 in the n = 2i + 1 dimensional case; Subsection 3.2 in the n = 2i ≡ 0 mod 4 dimensional case and in Subsection 3.3 the n = 2i ≡ 2 mod 4 dimensional case.…”

“…Hence, one cannot verify the Morse inequalities for abstract Lyapunov graphs, however, we can verify the Poincaré-Hopf inequalities. For more details see [1] and [2].…”

“…In [1] the Poincaré-Hopf inequalities in all generality are introduced for flows on isolating blocks N and their Lyapunov graph L N in order to ensure the continuation of L N to a Morse type Lyapunov graph. These inequalities involve the Betti numbers of the exiting and entering boundaries of N .…”

“…This system was defined in a more general setting by a continuation algorithm in [1] and appears in Section 2 as (2.6) and (2.7). In [1] it was shown that (2.6) and (2.7) have a nonnegative solution if and only if the Poincaré-Hopf inequalities (2.3), (2.4) and (2.5) are satisfied. This is the subject of Proposition 2.1 stated in Section 2.…”