In the geometric case, let us assume that $\psi$ is non-isotrivial ; if $F$ is a function field over an infinite field, let us moreover assume that it is a polynomial. The following are then equivalent :

1. (1)

   the heights $h_{\varphi}$ and $h_{\psi}$ coincide ;

2. (2)

   $\varphi$ and $\psi$ have infinitely many common preperiodic points ;

3. (3)

   the essential lowest bound of $h_{\varphi}+h_{\psi}$ is zero ;

4. (4)

   the equilibrium measures $\mu_{\varphi}$ and $\mu_{\psi}$ are equal at all places ;

5. (5)

   the metrized line bundles $\mathscr{O}(1)_{\varphi}$ and $\mathscr{O}(1)_{\psi}$ are isomorphic, up to a family of constants $(c_{v})$ such that $\prod c_{v}=1$.