Let us assume that $\overline{L}$ and $\overline{M}$ are the metrized line bundles $\overline{\mathscr{O}(1)}_{\varphi}$ and $\overline{\mathscr{O}(1)}_{\psi}$ attached to rational functions $\varphi$ and $\psi$ of degres $d$ and $e$ respectively, with $d\geq 2$ and $e\geq 2$. Let us write $h_{\varphi}$ and $h_{\psi}$ for the height relative to these metrized line bundles ; we call them the canonical heights. The isometry $\varphi^{*}\overline{\mathscr{O}(1)}_{\varphi}\simeq\overline{\mathscr{O}(1)}_{\varphi}^{d}$ and the functorial properties of the height imply that for any $x\in{\mathbf{P}}^{1}(\overline{F})$, $h_{\varphi}(\varphi(x))=dh_{\varphi}(x)$ and $h_{\psi}(\psi(x))=eh_{\psi}(x)$. In particular, preperiodic points for $\varphi$ (i.e., points with finite forward orbit) satisfy $h_{\varphi}(x)=0$. Moreover,

hence $({\widehat{c}}_{1}(\overline{\mathscr{O}(1)}_{\varphi})^{2}|{\mathbf{P}}^{1})=0$ since $d\neq 0,1$. Similarly, preperiodic points of $\psi$ satisfy $h_{\psi}(x)=0$, and $({\widehat{c}}_{1}(\overline{\mathscr{O}(1)}_{\psi})^{2}|{\mathbf{P}}^{1})=0$.

In the arithmetic case, or over function fields over a finit field, Northcott’s finiteness theorem implies easily that points $x$ such that $h_{\varphi}(x)=0$ are preperiodic for $\varphi$, and similarly for $\psi$. This is not true in general : for example, if $\varphi$ is constant, all constant points have height $0$ but only countably many of them are preperiodic ; more generally isotrivial rational functions, i.e. rational functions which are constant after conjugacy by an automorphism of ${\mathbf{P}}^{1}$ will furnish counterexamples. The best known result is restricted to (non-isotrivial) polynomials : by Benedetto [10], a point of height zero is then preperiodic ; the proof relies on a detailed analysis of the Julia set.

Let us show how Prop. 3.4.1 implies results of Baker and DeMarco [4], and of Petsche, Szpiro and Tucker [45].