Bulletin of the
Korean Mathematical Society
BKMS

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

Article

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Bull. Korean Math. Soc. 2018; 55(5): 1317-1332

Online first article September 10, 2018      Printed September 1, 2018

https://doi.org/10.4134/BKMS.b160064

Copyright © The Korean Mathematical Society.

Height inequality for rational maps and bounds for preperiodic points

Chong Gyu Lee

Soongsil University

Abstract

In this paper, we introduce the $D$-ratio of a rational map $f:\mathbb{P}^n$ -$\rightarrow \mathbb{P}^n$, defined over $\overline{\mathbb{Q}}$, whose indeterminacy locus is contained in a hyperplane $H$ on $\mathbb{P}^n$. The $D$-ratio $r(f;\overline{V})$ characterizes endomorphisms and provides a useful height inequality on $\mathbb{P}^n(\overline{\mathbb{Q}}) \setminus H$. We also provide a dynamical application: preperiodic points of dynamical systems of small $D$-ratio are of bounded height.

Keywords: height, rational map, preperiodic points, $D$-ratio

MSC numbers: Primary 11G50, 37P30; Secondary 14G50, 32H50, 37P05