Topological robustness of non-saddle sets
A invariant set \(X\) of a flow \(\varphi: M\times{\mathbb R} \rightarrow M\) is a non-saddle set provided every neighborhood \(U\) of \(X\) admits a neighborhood \(V\) of \(X\) such that for every \(x\in V\), \(\varphi (\{x\}\times [0, \infty)) \subset U\) or \(\varphi (\{x\}\times (-\infty, 0]) \subset V\). The authors introduce the notions of dynamical robustness and topological robustness for parametrized families \(\{\varphi_{\lambda}\}_{\lambda\in{\mathbb R}}\) of flows in a locally compact and locally connected metric space \(M\). Roughly speaking, the former means that a non-saddle set has a local continuation made of non-saddle sets, and the latter means that an isolated invariant set has a local continuation made of isolated invariant sets with the same shape type. Motivated by the fact that robustness holds for the property of being an attractor, the authors consider the problem: is every non-saddle set dynamically robust? In the first part of the paper, the authors give a simple example to give a negative answer to this question, and obtain two results concerning the problem: does dynamical robustness imply topological robustness? The first theorem states: If \(\{\varphi_{\lambda}\}_{\lambda\in{\mathbb R}}\) is a parametrized differentiable family of flows in an \(n\)-manifold \(M\), and if a connected isolated non-saddle set \(K_0\) for \(\varphi_0\) is dynamically robust, then \(K_0\) is topologically robust. For the second theorem the authors consider regularly dynamical robustness, which means that a regular non-saddle set has a local continuation made of regular non-saddle sets, where regularity is in the sense of [\textit{J. M. R. Sanjurjo}, Mathematika 47, No.~1--2, 299--305 (2000; Zbl 1128.37302)]. The second theorem states: If \(\{\varphi_{\lambda}\}_{\lambda\in{\mathbb R}}\) is a parametrized family of flows in a locally compact ANR \(M\), and if a connected isolated regular non-saddle set \(K_0\) for \(\varphi_0\) is regularly dynamically robust, then \(K_0\) is topologically robust. In the second part of the paper the authors study the robustness of dual sets, where duality is in the sense of [\textit{A. Giraldo, M. A. Morón, F. R. Ruiz del Portal} and \textit{J. M. R. Sanjurjo}, Topology Appl. 113, No.~1--3, 51--59 (2001; Zbl 1003.37009)]. The third theorem states: If \(\{\varphi_{\lambda}\}_{\lambda\in{\mathbb R}}\) is a parametrized family of flows in a compact metric space \(M\), and if a regular non-saddle connected set \(K_0\) for \(\varphi_0\) with connected and non-empty dual \(K_0'\) is regularly dynamically robust, then \(K_0'\) is also regularly dynamically robust for \(\varphi_0\).
scientific article; zbMATH DE number 5573804
scientific article