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The numbers of periodic orbits hidden at fixed points of holomorphic maps. (English) Zbl 07277636
Summary: Let \(f\) be an \(n\)-dimensional holomorphic map defined in a neighborhood of the origin such that the origin is an isolated fixed point of all of its iterates, and let \(\mathcal{N}_M(f)\) denote the number of periodic orbits of \(f\) of period \(M\) hidden at the origin. Gorbovickis gives an efficient way of computing \(\mathcal{N}_M(f)\) for a large class of holomorphic maps. Inspired by Gorbovickis’ work, we establish a similar method for computing \(\mathcal{N}_M(f)\) for a much larger class of holomorphic germs, in particular, having arbitrary Jordan matrices as their linear parts. Moreover, we also give another proof of the result of I. Gorbovickis [Bull. Sci. Math. 138, No. 3, 356–375 (2014; Zbl 1330.37046)] using our method.
MSC:
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics
37F46 Bifurcations; parameter spaces in holomorphic dynamics; the Mandelbrot and Multibrot sets
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
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