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We study zero entropy automorphisms of a compact K\"ahler manifold $X$. Ourgoal is to bring to light some new structures of the action on the cohomologyof $X$, in terms of the so-called dynamical filtrations on $H^{1,1}(X, {\mathbbR})$. Based on these filtrations, we obtain the first general upper bound onthe 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 $\ell_{\rmess}(G, X)$ for every zero entropy subgroup $G$, again in terms of thedimension of $X$ only. We propose a conjectural upper bound for the essentialnilpotency class $c_{\rm ess}(G,X)$ of a zero entropy subgroup $G$.  Finally, we construct examples showing that our upper bound of the polynomialgrowth (as well as the conjectural upper bound of $c_{\rm ess}(G,X)$) areoptimal.

Zero entropy automorphisms of compact K\"ahler manifolds and dynamical filtrations