  This paper develops further the theory of the automorphic group of
non-constant entire functions. This theory essentially started with two
remarkable papers of Tatsujir\^o Shimizu that were published in 1931. There are
three results in this paper. The first result is that the ${\rm Aut}(f)$-orbit
of any complex number has no finite accumulation point. The second result is an
accurate computation of the derivative of an automorphic function of an entire
function at any of its fixed points. The third result gives the precise form of
an automorphic function that is uniform over an open subset of $\mathbb{C}$.
This last result is a follow up of a remarkable theorem of Shimizu. It is a
sharp form of his result. It leads to an algorithm of computing the entire
automorphic functions of entire functions. The complexity is computed using an
height estimate of a rational parameter discovered by Shimizu.
