Stable sets of planar homeomorphisms with translation pseudo-arcs (Q2305703)

The article indicates how complicated the topological structure of stable (or unstable) sets may be. The main theorem constructs a planar homeomorphism \(f\) whose unique compact invariant set is the origin \(\{0\}\) and such that the branches of the stable (\(S\)) and unstable (\(U\)) sets of \(\{0\}\) (connected components of \(S\setminus\{0\}\) and \(U\setminus\{0\}\)) admit translation pseudo-arcs. This means that every connected component of \(S\setminus\{0\}\) (or \(U\setminus\{0\}\)) is equal to the orbit under \(f\) of a pseudo-arc \(K\) such that \(f(K) \cap K\) is a singleton. The construction is first done in the case for which \(S\) and \(U\) possess only one branch and is later extended to any number of branches (equal for \(S\) and \(U\) and alternating around \(\{0\}\)). The technique used transforms an invariant straight line into a translation pseudo-arc with an arbitrarily small perturbation.