  We study zero entropy automorphisms of a compact K\"ahler manifold $X$. Our
goal is to bring to light some new structures of the action on the cohomology
of $X$, in terms of the so-called dynamical filtrations on $H^{1,1}(X, {\mathbb
R})$. Based on these filtrations, we obtain the first general upper bound on
the polynomial growth of the iterations $(g^m)^* \, {\circlearrowleft} \,
H^2(X, {\mathbb C})$ where $g$ is a zero entropy automorphism, in terms of
${\rm dim} \, X$ only.
  We also give an upper bound for the (essential) derived length $\ell_{\rm
ess}(G, X)$ for every zero entropy subgroup $G$, again in terms of the
dimension of $X$ only. We propose a conjectural upper bound for the essential
nilpotency class $c_{\rm ess}(G,X)$ of a zero entropy subgroup $G$.
  Finally, we construct examples showing that our upper bound of the polynomial
growth (as well as the conjectural upper bound of $c_{\rm ess}(G,X)$) are
optimal.
