Generic Bowen-expansive flows
The paper examines, in a generic sense, the role of expansiveness for \(C^1\) flows on a finite-dimensional closed Riemannian manifold \(M\) with metric \(d\). Let \({\mathfrak X}^1(M)\) denote the set of \(C^1\) vector fields on \(M\), let \(X_t\) the flow generated by \(X\), and let \({\mathcal B}\) be the set of increasing homeomorphisms of \({\mathbb R}\). A flow \(X_t\) is Bowen-expansive if for every \(\epsilon>0\) there is \(\delta>0\) such that if \(x,y\in M\) and for some \(h\in{\mathcal B}\) we have \(d(X_t(x),X_{h(t)}(y))<\delta\) for all \(t\in{\mathbb R}\), then \(y\in \{ X_t(x):-\epsilon\leq t\leq \epsilon\}\). The main result is the existence of a residual subset \({\mathcal R}\) of \({\mathfrak X}^1(M)\) such that if \(X\in {\mathcal R}\) is Bowen-expansive, then \(X\) is Axiom A without cycles. The proof uses similar arguments found in the proof of the analogous result for expansive diffeomorphisms.
scientific article; zbMATH DE number 6136414
scientific article