The Enriques surface of minimal entropy

TL;DR

This paper investigates the realization of Lehmer's number as a dynamical degree of automorphisms on Enriques surfaces, showing it cannot occur in odd characteristic but does in characteristic 2, with explicit examples provided.

Contribution

It extends known results about Lehmer's number to Enriques surfaces in positive characteristic and constructs explicit automorphisms realizing this number in characteristic 2.

Findings

Lehmer's number cannot be realized by automorphisms of Enriques surfaces in odd characteristic.

In characteristic 2, a unique Enriques surface admits an automorphism with dynamical degree Lehmer's number.

Explicit equations for the surface and automorphisms realizing Lehmer's number are provided.

Abstract

Lehmer's number $\lambda_{10}$ is the smallest dynamical degree greater than $1$ that can occur for an automorphism of an algebraic surface. We show that $\lambda_{10}$ cannot be realized by automorphisms of Enriques surfaces in odd characteristic, extending a result of Oguiso over the complex numbers. In contrast, we prove that in characteristic $2$ there exists a unique Enriques surface that admits an automorphism with dynamical degree $\lambda_{10}$. We also provide explicit equations for the surface as well as for all conjugacy classes of automorphisms that realize $\lambda_{10}$.