Links of homeomorphisms of surfaces and topological entropy
Let f be an orientation preserving self-homeomorphism of a compact surface F with an f-invariant set \(\Sigma\) of finite points in Int F. Let \(M_ f\) be the mapping torus of f. Then \(M_ f\) is a 3-dimensional manifold, and we can naturally construct a link \(L_{f,\Sigma}\) in \(M_ f\) from \(\Sigma\). In this paper, we show that if the topological entropy h(f) of f is equal to zero, then \(L_{f,\Sigma}\) is a graph link. And, conversely, if \(L_{f,\Sigma}\) is a graph link, then f is isotopic rel \(\Sigma\) to a homeomorphism g such that \(h(g)=0\). Moreover, if \(L_{f,\Sigma}\) is not a graph link and f is differentiable at each point of \(\Sigma\), then f has infinitely many periodic orbits whose periods are mutually distinct.
scientific article; zbMATH DE number 3905508