Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2023; 60(3): 659-675

Online first article May 16, 2023      Printed May 31, 2023

https://doi.org/10.4134/BKMS.b220292

Copyright © The Korean Mathematical Society.

The critical pods of planar quadratic polynomial maps of topological degree 2

Misong Chang, Sunyang Ko, Chong Gyu Lee, Sang-Min Lee

Soongsil University; Soongsil University; Soongsil University; Soongsil University

Abstract

{Let $K$ be an algebraically closed field of characteristic 0 and let $f$ be a non-fibered planar quadratic polynomial map of topological degree 2 defined over $K$. We assume further that the meromorphic extension of $f$ on the projective plane has the unique indeterminacy point.} We define \emph{the critical pod of $f$} where $f$ sends a critical point to another critical point. By observing the behavior of $f$ at the critical pod, we can determine a good conjugate of $f$ which shows its statue in GIT sense.

Keywords: Critical pod, quadratic polynomial map, moduli space, conjugacy class, dynamical Mordell-Lang conjecture, semistable maps, critical point, fixed point

MSC numbers: Primary 37P05, 37F12; Secondary 12E12

Supported by: This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (NRF-2016R1D1A1B01009208 and NRF-2021R1A6A1A10044154).