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The numbers of periodic orbits hidden at fixed points of holomorphic maps
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 41 / Issue 2 / February 2021
- Published online by Cambridge University Press:
- 03 September 2019, pp. 578-592
- Print publication:
- February 2021
-
- Article
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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.
