Height growth under a morphism of the projective line
= Height growth under a morphism of the projective line
If $F:\mathbb P^1\to\mathbb P^1$ is a morphism of degree $d$ over $\mathbb Q$, there are positive constants $c_1,c_2$ such that
$$
c_1H(P)^d\leq H(F(P))\leq c_2H(P)^d
$$
for every $P\in\mathbb P^1(\mathbb Q)$.