Height inequality for rational maps and bounds for preperiodic points
Bull. Korean Math. Soc. 2018 Vol. 55, No. 5, 1317-1332
Published online September 1, 2018
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
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