On automorphisms of Enriques surfaces and their entropy

TL;DR

This paper investigates automorphisms of Enriques surfaces, establishing bounds on their lifts to K3 covers, deriving constraints on their dynamical degrees, and identifying potential minimal Salem numbers relevant to their dynamics.

Contribution

It provides new bounds on the order of automorphism lifts, introduces mod 2 constraints on automorphisms, and classifies potential minimal Salem numbers for Enriques surfaces.

Findings

Bound on the order of the lift acting on anti-invariant cohomology

Mod 2 constraints on automorphisms of Enriques surfaces

Complete list of potential minimal Salem numbers

Abstract

Consider an arbitrary automorphism of an Enriques surface with its lift to the covering K3 surface. We prove a bound of the order of the lift acting on the anti-invariant cohomology sublattice of the Enriques involution. We use it to obtain some mod 2 constraint on the original automorphism. As an application, we give a necessary condition for Salem numbers to be dynamical degrees on Enriques surfaces and obtain a new lower bound on the minimal value. In the Appendix, we give a complete list of Salem numbers that potentially may be the minimal dynamical degree on Enriques surfaces and for which the existence of geometric automorphisms is unknown.