Chain recurrence, semiflows, and gradients
Let \((X, d)\) be a metric space. A semiflow is a continuous map \(\phi: [0, \infty)\times X\to X\) satisfying (i) \(\forall x\in X\); \(\phi(0, x)= x\), and (ii) \(\forall s, t\geq 0 \forall x\in X\); \(\phi(s, \phi(t, x))= \phi(s+ t, x)\). The time-one map of \(\phi\) is denoted by \(h: h(x)= \phi(1, x)\). In the case of \(X\) compact, \textit{C. Conley} [Isolated invariant sets and the Morse index, Am. Math. Soc., Providence, Rhode Island (1978; Zbl 0397.34056)] gave a characterization of the chain recurrent set \({\mathcal C}{\mathcal R}(\phi)\) of \(\phi\) in terms of attractors of \(\phi\). Namely, he proved that \(X\backslash {\mathcal C}{\mathcal R}(\phi)= \bigcup[B(A)\backslash A]\) as \(A\) varies over all attractors of \(\phi\) and where \(B(A)\) denotes the basin of attraction of \(A\). In the present paper, the author extends that result to the case of an arbitrary metric space. Moreover, the author shows that the chain recurrent set of \(\phi\) is the same as the chain recurrent set of its time-one map \(h\). Applying these results, the author establishes some conditions on a real- valued function which ensure that the semiflow generated by its gradient has only equilibria in its chain recurrent set.
scientific article; zbMATH DE number 800417
scientific article