Fixed point indices of iterations of planar homeomorphisms
Let us consider a local homeomorphism \(f : U \to \mathbb{R}^2\) \((U\) is an open neighborhood of 0) such that for each \(n\neq 0\) the origin 0 is an isolated fixed-point for \(f^n\). For natural \(n\), define integers \(i_n(f,0)\) by the equality: \(i_n(f,0)=\sum_{k| n}\mu(n/k)\text{ind}(f^k,0)=\sum_{k| n}\mu(k)\text{ind}(f^{n/k},0)\), where \(\mu\) is the classisal arithmetical Möbius function. The authors define the set \(A\) of generators for \(\{\text{ind}(f^n,0)\}^\infty_{n=1}\) as follows: \(A=\{a\in \mathbb{N}: \text{ind}(f^a,0)= p\) and for all \(b| a\) such that \(b\neq a\) \(\text{ind}(f^b,0) = 1\}\) and the following result is proved: Let \(f\) be an orientation-preserving local homeomorphism of the plane, \(A\neq\emptyset\). Then \(A\) is finite and LCM\((A)|(p-1)\), where LCM\((A)\) denotes the lowest common multiple of all elements in \(A\). Further, similar results concern orientation-reversing local homeomorphims. The paper is well written and organized.
scientific article; zbMATH DE number 2096722