  We study the automorphisms of compact K\"ahler manifolds having slow
dynamics. By adapting Gromov's classical argument, we give an upper bound on
the polynomial entropy and study its possible values in dimensions $2$ and $3$.
We prove that every automorphism with sublinear derivative growth is an
isometry ; a counter-example is given in the $C^{\infty}$ context, answering
negatively a question of Artigue, Carrasco-Olivera and Monteverde on polynomial
entropy. Finally, we classify minimal automorphisms in dimension $2$ and prove
they exist only on tori. We conjecture that this is true for any dimension.
