zbMATH — the first resource for mathematics

On automorphisms of Enriques surfaces and their entropy. (English) Zbl 1401.14171
Let \(S\) be an Enriques surface and let \(\tilde{S}\) be its universal cover, which is a K3 surface. Let \(\epsilon: \tilde{S}\to S\) be the covering involution.
Let \(f\in \mathrm{Aut}(S)\) be an automorphism. Then \(f\) admits a (nonunique) lift to \(\tilde{S}\). Fix such a lift \( \tilde{f}\). In this paper the authors study the action of \(\tilde{f}\) on the orthogonal complement of \(H^2(\tilde{S},\mathbb{Z})^{\epsilon}\) and deduce various property of \(f\) from this.
First the authors show that \(f\) has finite order, and find several bounds on the order. This leaves 31 possibilities for the order.
Then they study the dynamical degree \(\lambda(f)\) of the automorphism \(f\), which is known to be a Salem number. The authors consider the minimal polynomial of the algebraic integer \(\lambda(f)\) and show that when this polynomial is reduced modulo 2 then it a product of several \(m\)-th cyclotomic polynomials with \(m\in \{1,3,5,7,9,15\}\).
In the appendix to the paper the authors give a complete list of possible dynamical degrees and indicate for each of them whether an example has been constructed or not.

14J28 \(K3\) surfaces and Enriques surfaces
14J50 Automorphisms of surfaces and higher-dimensional varieties
37B40 Topological entropy
Full Text: DOI