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Let F and G be morphisms of degree at least 2 from P^N to P^N that aredefined over the algebraic closure of Q. We define the arithmetic distanced(F,G) between F and G to be the supremum over all algebraic points P of|h_F(P)-h_G(P)|, where h_F and h_G are the canonical heights associated to themorphisms F and G, respectively. We prove comparison theorems relating d(F,G)to more naive height functions and show that for a fixed G, the set of Fsatisfying d(F,G) < B is a set of bounded height. In particular, there are onlyfinitely many such F defined over any given number field.

Canonical heights and the arithmetic complexity of morphisms on projective space