Canonical height

Remarks

The limit exists because $h(\phi(Q)) = d \, h(Q) + O(1)$ uniformly in $Q$, so the sequence $d^{-n} h(\phi^n(P))$ is Cauchy, by a telescoping argument. It has these properties, due to Call and Silverman:

1. $\hat{h}_\phi(\phi(P)) = d \, \hat{h}_\phi(P)$;
2. $\hat{h}_\phi - h$ is bounded on $\mathbb{P}^N(\overline{\Q})$;
3. $\hat{h}_\phi(P) \ge 0$, with equality exactly when $P$ is preperiodic for $\phi$.

Property 3 uses Northcott's theorem: a point of canonical height $0$ has an orbit of bounded height inside a fixed number field, and such an orbit is finite.

Example: for $\phi(z) = z^2$ on $\mathbb{P}^1$, $h(\phi(P)) = 2 h(P)$ exactly, so $\hat{h}_\phi = h$. Its zeros are $0$, $\infty$ and the roots of unity, which are exactly the preperiodic points of $z \mapsto z^2$.

Non-example: for $\psi(z) = z^2 + 1$ the naive height $h$ is not the canonical height of $\psi$. It fails property 1: $h(1) = 0$, but $h(\psi(1)) = h(2) = \log 2 \ne 2 h(1)$.

For an elliptic curve $E$ over a number field, the Lattès map that describes doubling on $x$-coordinates has degree $4$. Its canonical height, up to a normalizing factor, is the Néron–Tate height of $E$.