AbstractWe study the minimally displaced set of irreducible automorphisms of a free group. Our main result is the co-compactness of the minimally displaced set of an irreducible automorphism with exponential growth $$\phi $$
ϕ
, under the action of the centraliser $$C(\phi )$$
C
(
ϕ
)
. As a corollary, we get that the same holds for the action of $$ <\phi>$$
<
ϕ
>
on $$Min(\phi )$$
M
i
n
(
ϕ
)
. Finally, we prove that the minimally displaced set of an irreducible automorphism of growth rate one consists of a single point.