Zero entropy automorphisms of compact Kähler manifolds and dynamical filtrations

Abstract

We study zero entropy automorphisms of a compact Kähler 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 $$\dim X$$ only. We also give an upper bound for the (essential) derived length $$\ell _{\mathrm{ess}}(G, X)$$ 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 $$c_\mathrm{ess}(G,X)$$ 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 $$c_\mathrm{ess}(G,X)$$ ) are optimal.