Closures of asymptotic curves in a two-dimensional symplectic map
Let \(f\) be a \(C^1\)-smooth area-preserving diffeomorphism of a 2D manifold \(M\) and \(z^*\in M\) a hyperbolic fixed point of the map \(f\). Such point \(z^*\in M\) generates four asymptotic curves. Following the author we call these curves separatrix branches and denote the stable branches by \(\Gamma_{1,2}^s\) and the unstable ones by \(\Gamma_{1,2}^u\). The author deals with the following questions: Suppose that (a) these branches lie in a compact invariant domain, (b) these branches intersect at some point \(w^*\), \(w^*\neq z^*\). Do the closures of these curves coincide? The author gives a positive answer to this question.
scientific article; zbMATH DE number 1431864