  We address the following open problem, implicit in the 1990 article
"Automorphisms of one-sided subshifts of finite type" of Boyle, Franks and
Kitchens (BFK):
  "Does there exists an element $\psi$ in the group of automorphisms of the
one-sided shift $\operatorname{Aut}(\{0,1,\ldots,n-1\}^{\mathbb{N}},
\sigma_{n})$ so that all points of $\{0,1,\ldots,n-1\}^{\mathbb{N}}$ have
orbits of length $n$ under $\psi$ and $\psi$ is not conjugate to a
permutation?"
  Here, by a 'permutation' we mean an automorphism of one-sided shift dynamical
system induced by a permutation of the symbol set $\{0,1,\ldots,n-1\}$.
  We resolve this question by showing that any $\psi$ with properties as above
must be conjugate to a permutation.
  Our techniques naturally extend those of BFK using the strongly synchronizing
automata technology developed here and in several articles of the authors and
collaborators (although, this article has been written to be largely
self-contained).
