Topological equivalence of diffeomorphisms and curves
We are given a compact 2-dimensional \(C^r\)-manifold \(S\), \(p \in S\), and \(\text{Diff}^r(S,p)\), which consists of all \(C^r\)-diffeomorphisms \(\phi : S \to S\) with \(\phi(p) = p\). Take \(I^r(p)\) to be the set of all (reviewer assumes \(C^r\)-) imbeddings \(\alpha : I = (-\varepsilon, \varepsilon) \to S\) with \(\alpha (0) = p\) (we take it that \(\varepsilon\) is a fixed positive number, but this is never stated). Then \(G^r(p) = \text{Diff}^r(S,p) \times I^r(p)\) is given the \(C^r\) product topology. One defines \((\phi, \alpha)\), \((\phi', \alpha') \in G^r(p)\) to be \(C^0\)-conjugate if there is a \(C^0\)-conjugacy \(h : (S,p) \to (S,p)\) between \(\phi\) and \(\phi'\) such that \(h(\alpha(I)) \subset \alpha'(I)\). We presume from the authors' hint that an element \((\phi,\alpha) \in G^r(p)\) is structurally stable in \(G^r(p)\) if it has a neighborhood in \(G^r(p)\) consisting of \(C^0\)-conjugates. Three ``essentially different generic cases'' are considered for \((\phi, \alpha) \in G^r(p):\) nodal, saddle, and focal. These are based on the nature of the eigenvalues of \(d\phi(p)\) and some other positional requirements relative to \(\alpha(I)\) and, for example, the eigenspaces. There are three major theorems provided. The simplest one is in the nodal case (Theorem A) where it is shown that all such \((\phi, \alpha)\) are structurally stable in \(G^r(p)\). The other cases are more complicated and are described in terms of a certain foliation by codimension one submanifolds of \(G^r(p)\), and what exists on the leaves of this foliation. The main complication in this theory arises when \(\phi\) has eigenvalues which are neither real nor purely imaginary and whose arguments are irrational.
scientific article; zbMATH DE number 759471