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The numbers of periodic orbits hidden at fixed points of holomorphic maps

Part of: Smooth dynamical systems: general theory Holomorphic mappings and correspondences Complex dynamical systems

Published online by Cambridge University Press:  03 September 2019

JIANYONG QIAO ,
HONGYU QU  and
GUANGYUAN ZHANG
JIANYONG QIAO
Affiliation:
School of Sciences, Beijing University of Posts and Telecommunications, Beijing100786, PR China email qjy@bupt.edu.cn
HONGYU QU
Affiliation:
School of Computer Science, Beijing University of Posts and Telecommunications, Beijing100786, PR China email hongyuqu@bupt.edu.cn
GUANGYUAN ZHANG
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing100084, PR China email gyzhang@math.tsinghua.edu.cn
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Abstract

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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 Gorbovickis [On multi-dimensional Fatou bifurcation. Bull. Sci. Math.138(3)(2014) 356–375] using our method.

Keywords

smooth dynamicsperiodic orbitsholomorphic mapsfixed-point indicesDold indices

MSC classification

Primary: 32H50: Iteration problems 37C25: Fixed points, periodic points, fixed-point index theory
Secondary: 37F45: Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations 37F50: Small divisors, rotation domains and linearization; Fatou and Julia sets
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 2 , February 2021 , pp. 578 - 592
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Copyright
© Cambridge University Press, 2019

References

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