  The classical result by Dyer--Scott about fixed subgroups of finite order
automorphisms of $F_n$ being free factors of $F_n$ is no longer true in
$Z^m\times F_n$. Within this more general context, we prove a relaxed version
in the spirit of Bestvina--Handel Theorem: the rank of fixed subgroups of
finite order automorphisms is uniformly bounded in terms of $m,n$. We also
study periodic points of endomorphisms of $Z^m\times F_n$, and give an
algorithm to compute auto-fixed closures of finitely generated subgroups of
$Z^m\times F_n$. On the way, we prove the analog of Day's Theorem for real
elements in $Z^m\times F_n$, contributing a modest step into the project of
doing so for any right angled Artin group (as McCool did with respect to
Whitehead's Theorem in the free context).
