A dynamical Shafarevich theorem for endomorphisms of ${\mathbb{P}}^N$

We prove a dynamical analogue of the Shafarevich conjecture for morphisms $f:{\mathbb{P}}_K^N\to {\mathbb{P}}_K^N$ of degree $d\ge 2$, defined over a number field $K$. This extends previous work of Silverman and others in the case $N=1$.