Stability concepts in topological dynamics by nonstandard methods (Q1376324)

Motivated by, and using, methods of nonstandard analysis, the paper introduces four topologically-defined concepts of stability of a continuous semiflow on a compact subset \(K\) of \(R^{n}\). Standard versions of the results are also presented. Unfortunately, the paper lacks examples showing that the stability definitions it presents are interesting. The set of semiflows is topologized by \(d(f,g) = \sup_{x\in K, 0\leq t\leq 1}|f(t,x) - g(t,x)|\). The first (and most restrictive) of the definitions states that a flow \(f\) is strongly s-stable if for any \(\varepsilon > 0\) one can choose a neighborhood \(G\) of \(f\) such that for any \(g\in G\), \(|f(t,x) - g(t,x)|< \varepsilon \) for all \(t\) and \(x\). It seems to the reviewer that if the omega limit set for \(f\) of some point consists of more than one point, then defining \(g(t,x) \equiv f(kt, x)\) for some \(k\) near 1 shows that \(f\) is not strongly s-stable. The second stability definition (`deformational stability') requires that the semiflow have the property that \(q\) lie on the closure of the forward orbit of \(p\) whenever it is possible to find an \(\varepsilon\)-chain from \(p\) to \(q\) for each \(\varepsilon >0\). It is shown that any minimal flow has this property; for flows it seems clear that if the phase space is connected then the converse also holds. (On a connected phase space, if there is any ordered pair of points \((p,q)\) and \(\varepsilon >0\) where it is not possible to find an \(\varepsilon\)-chain from \(p\) to \(q\), then there is a nontrivial attractor-repellor pair \((A,A*)\) with \(A\), \(A*\) nonempty disjoint, invariant, and there are \(\varepsilon\)-chains going from \(A*\) to \(A\).) The third stability concept is related to the author's version of the pseudo-orbit tracing property (POTP). Both of these definitions suffer from the failure to consider the necessity of reparameterization of the \(t\) coordinates. The final stability concept is approximate stability. Roughly speaking, a flow \(f\) is approximately stable if it has neighborhoods \(G\) such that for any two flows \(g_1\), \(g_2\) in \(G\), any point that is approximately nonwandering for one is approximately nonwandering for the other. (Here \(p\) is `approximately nonwandering' if some point in a small neighborhood of \(p\) returns to that neighborhood after a sufficiently long time; the neighborhood \(G\) depends on the length of time required and the neighborhood of \(p\).) It is shown that a minimal flow has this property.