The Enriques surface of minimal entropy

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 ${\lambda }_{10}$.