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Virtually repelling fixed points. (English) Zbl 1043.37014

The paper deals with the study of virtually repelling fixed-points of the holomorphic germs \(f: (\mathbb{C},\alpha)\to (\mathbb{C},\alpha)\) for which \(f(\alpha)=\alpha\). The multiplicity \(m\) of \(\alpha\) as a fixed-point of \(f\) is the multiplicity of \(\alpha\) as a root of \(z-f(z)\). Then the fixed-point \(\alpha\) is virtually repelling if \(\text{Re}[\text{Res}_{z=\alpha}(z- f(z))^{-1}]< m/2\). Several results on virtually repelling fixed-points are obtained, e.g., that a germ \(f: (\mathbb{C},\alpha)\to (\mathbb{C},\alpha)\) has a virtually repelling fixed-point at \(\alpha\) iff any sufficiently small perturbation \(f_\varepsilon\) of \(f\) has at least one virtually repelling fixed point close to \(\alpha\). Also, the author proves that most proper holomorphic mappings \(f: U\to V\) with \(U\) contained in \(V\) have at least one virtually repelling fixed-point.

MSC:

37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
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