  A Houghton's group $\mathcal{H}_n$ consists of translations at infinity of a
$n$ rays of discrete points on the plane. In this paper we study the growth
rate of endomorphisms of Houghton's groups. We show that if the kernel of an
endomorphism $\phi$ is not trivial then the growth rate $\mathrm{GR}(\phi)$
equals either $1$ or the spectral radius of the induced map on the
abelianization. It turns out that every monomorphism $\phi$ of $\mathcal{H}_n$
determines a unique natural number $\ell$ such that $\phi(\mathcal{H}_n)$ is
generated by translations with the same translation length $\ell$. We use this
to show that $\mathrm{GR}(\phi)$ of a monomorphism $\phi$ of $\mathcal{H}_n$ is
precisely $\ell$ for all $2\leq n$.
