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Nonsingular Smale flows on \(S^ 3\). (English) Zbl 0609.58039

From the introduction: ”The qualitative description of the dynamical behavior of a smooth flow is often divided into two parts: the ”gradient- like” behavior reflected in the existence of a Lyapunov function, and the ”chain recurrent” behavior. In studying a class of dynamical systems it is important to understand two things. First, for each of these two parts what kinds of behavior are possible, and second, how do these two aspects of dynamical behavior interact?”
This paper deals with flows without singular points, Lyapunov functions and a finite, connected, oriented graph associated with them, which possesses no oriented cycles, and each vertex of which is labelled with a chain recurrent flow on a compact space, named Lyapunov graph. The main result is the following existence theorem: Suppose \(\Gamma\) is an (abstract) Lyapunov graph whose sinks and sources are each labelled with a single periodic orbit, and suppose each remaining vertex is labelled with the suspension of a subshift of finite type. Then \(\Gamma\) is associated with a non-singular Smale flow \(\phi_ t\) and a Lyapunov function f on \(S^ 3\), if and only if the following is satisfied: (1) The graph \(\Gamma\) is a tree, with one edge attached to each source and sink vertex. (2) Suppose v is any other vertex labelled with the subshift suspension \(\sigma(A)\) and with \(e^+\) entering edges and \(e^-\) exiting edges. If \(k=\dim \ker ((I-\bar A): F^ n_ 2\to F^ n_ 2)\) where \(F_ 2=Z/2\) and \(\bar A\) is the mod 2 reduction of A, then \(e^+_ v\leq k_ v+1\), \(e^-_ v\leq k+1\), and \(k_ v+1\leq d^+_ v+e^-_ v\).

MSC:

37C10 Dynamics induced by flows and semiflows
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