We consider approximation rates of sparsely connected deep rectified linear unit (ReLU) and rectified power unit (RePU) neural networks for functions in Besov spaces $B^α_{q}(L^p)$ in arbitrary dimension $d$, on general domains. We show that \alert{deep rectifier} networks with a fixed activation function attain optimal or near to optimal approximation rates for functions in the Besov space $B^α_τ(L^τ)$ on the critical embedding line $1/τ=α/d+1/p$ for \emph{arbitrary} smoothness order $α>0$. Using interpolation theory, this implies that the entire range of smoothness classes at or above the critical line is (near to) optimally approximated by deep ReLU/RePU networks.