Pseudo-orbit tracing property and strong transversality of diffeomorphisms on closed manifolds
Let \(f\) be a \(C^ 1\) diffeomorphism of a closed manifold \(M\) with a metric \(d\). In the paper the dynamical properties of \(f\) having the pseudo-orbit tracing property (abbrev. POTP) are investigated. A sequence \(\{x_ i\}_{i \in \mathbb{Z}}\), \(x_ i \in M\), is called \(\delta\)-pseudo-orbit of \(f\) if \(d(f(x_ i); x_{i+1}) < \delta\), \(i \in \mathbb{Z}\). A sequence \(\{x_ i\}_{i \in \mathbb{Z}}\), \(x_ i \in M\), is said to be \(f\)-\(\varepsilon\)-traced by \(y \in M\) if \(d(f^ i(y),x_ i) < \varepsilon\), \(i \in \mathbb{Z}\). Definition. A diffeomorphism \(f\) is said to have POTP if for every \(\varepsilon > 0\) there is \(\delta > 0\) such that every \(\delta\)-pseudo- orbit of \(f\) can be \(f\)-\(\varepsilon\)-traced by some \(y \in M\). \textit{C. Robinson} [Rocky Mt. J. Math. 7, 425-437 (1977; Zbl 0375.58016)] proved that every Axiom A diffeomorphism satisfying the strong transversality condition has POTP. The aim of the paper is to analyze whether POTP implies Axiom A and strong transversality. Partial results for this problem were obtained by \textit{K. Moriyasu} [Nagoya Math. J. 123, 91-102 (1991; Zbl 0748.58023)] for \(\dim M = 1\) and by the author [Nagoya Math. J. 126, 125-140 (1992; Zbl 0774.58032)] for \(\dim M = 3\). The main result of the paper is the following Theorem. The \(C^ 1\) interior of all diffeomorphisms having POTP of a closed manifold \(M\) coincides with the set of all Axiom A diffeomorphisms satisfying the strong transversality condition.
scientific article; zbMATH DE number 720170