Realization of all Dold's congruences with stability
This paper is concerned with sequences of integers satisfying Dold's congruences. In particular, it deals with the sequences of fixed point indices, at an isolated stable fixed point, of the iterations of a orientation-preserving homeomorphisms. The main result states that for each \(n>2\) and for every sequence \(\{I_m\}_{m\in{\mathbb N}}\) satisfying Dold's congruences there exists an orientation preserving homeomorphism \(h:{\mathbb R}^n\to{\mathbb R}^n\) such that the fixed point and the periodic point sets of \(h\) coincide with the origin \(\bar 0\) of \({\mathbb R}^n\), which is a stable equilibrium of \(h\), and the fixed point index \(i(h^m,\bar 0)\) at \(\bar 0\) is given by \(I_m\) for any \(m\in{\mathbb N}\). This fact can be used to deduce that in a dimension \(n> 2\) not all sequences are realized (in the above sense) by homeomorphisms and can also be realized by diffeomorphisms.
scientific article; zbMATH DE number 5759048
scientific article