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

TL;DR

This paper investigates zero entropy automorphisms of compact Kähler manifolds, revealing new cohomological structures and establishing bounds on their dynamical complexity, with optimal examples demonstrating the sharpness of these bounds.

Contribution

It introduces dynamical filtrations on cohomology, provides the first general upper bounds on polynomial growth and derived length for zero entropy automorphisms, and proposes a conjecture on nilpotency class.

Findings

First general upper bound on polynomial growth in terms of manifold dimension.

Upper bound on the essential derived length for zero entropy groups.

Construction of examples showing bounds are optimal.

Abstract

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.