On the fine structure of the global attractor of a uniformly persistent flow (Q413447)

Let \(E\) be a closed subset of a locally compact metric space \(X\) with boundary \(\partial E\). The dissipative flow \(\phi: E\times {\mathbb R} \to E\) (i.e., \(\omega(x)\neq \emptyset\) for all \(x\in E\) with \(\bigcup_{x\in E}\omega(x)\) having compact closure) is said to be uniformly persistent if there exists \(\beta>0\) such that, for every \(x\in E\), NEWLINE\[NEWLINE \liminf\{ d(\phi(x,t), \partial E): t\to \infty \} \geq \beta .NEWLINE\]NEWLINE If \(E\) is compact, then \(\phi\) is uniformly persistent if and only if \(\partial E\) is a repeller of \(\phi\) [\textit{B. M. Gray}, J. Math. Anal. Appl. 139, No. 2, 372--381 (1989; Zbl 0677.54033)].NEWLINENEWLINEThe author studies the internal structure of the global attractor of a uniformly persistent flow. It is shown that the existence of a repelling point \(p\) implies the existence of a dual attractor \(A\) with spherical shape (in the sense of [\textit{K. Borsuk}, Theory of shape. Warszawa: PWN -- Polish Scientific Publishers (1975; Zbl 0317.55006)]), whose basin of attraction is the interior of \({\mathbb R}^n_+ - \{ p \}\). Some natural Morse decompositions of uniformly persistent flows are also studied and their Morse equations are given, providing necessary and sufficient conditions for the existence of attractors with the shape of \({\mathbb S}^1\), or such that their suspension has spherical shape.