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Fiber Julia sets of polynomial skew products with super-saddle fixed points


Author: Shizuo Nakane
Journal: Proc. Amer. Math. Soc. 149 (2021), 2539-2550
MSC (2020): Primary 32H50, 37F10, 37C29
DOI: https://doi.org/10.1090/proc/15345
Published electronically: March 16, 2021
MathSciNet review: 4246804
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Abstract: If a polynomial skew product on $\mathbb {C}^2$ has a relation between two saddle fixed points, fiber Julia sets $J_z$ behave discontinuously. That is, as the base variable $z$ tends to a point $\beta$ corresponding to a saddle point, the limits of $J_z$ strictly include $J_{\beta }$. When the map is linearizable at these saddle points, we have described their behaviors in terms of Lavaurs maps in [Indiana Univ. Math. J. 68 (2019), pp. 35–61]. In this article, we consider the case when the map is not invertible at a saddle fixed point. It turns out that the Lavaurs map must be identically zero. As a result, the limits of fiber Julia sets have non-empty interiors.


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Additional Information

Shizuo Nakane
Affiliation: Department of Mathematics, Tokyo Polytechnic University, 1583, Iiyama, Atsugi-city, Kanagawa 243-0297, Japan
MR Author ID: 190353
Email: nakane@gen.t-kougei.ac.jp

Received by editor(s): January 2, 2020
Received by editor(s) in revised form: April 20, 2020, and September 21, 2020
Published electronically: March 16, 2021
Additional Notes: The author was partially supported by JSPS KAKENHI Grant Number 16K05213.
Communicated by: Filippo Bracci
Article copyright: © Copyright 2021 American Mathematical Society