Local indices of iterations of a holomorphic map
Let \(f: (U,0)\to ({\mathbb R}^ m,0)\) be a \(C^ 1\) map of a neighborhood \(U\) of \(0\) such that \(0\) is an isolated fixed point for any iteration of \(f\). By results of Zabreĭko and Krasnosel'skii, Dold, and Shub and Sullivan, there is a (finite) sequence of integer numbers \(A_ d\) such that the indices of iterations of \(f\) at \(0\) can be written \(i(f^ n,0)=\sum_{d| n}dA_ d\), \(n=1,2,... \). In this paper the author shows that if \(f\) is complex analytic (then \(m=2k\) and \({\mathbb R}^ m={\mathbb C}^ k)\) then the multiplicities \(A_ d\) are submitted to some restrictions. A complete discussion is given by the author only in the \({\mathbb C}^ 2\) case.
scientific article; zbMATH DE number 4101570
scientific article