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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.

##### MSC:
 14J28 $$K3$$ surfaces and Enriques surfaces 14J50 Automorphisms of surfaces and higher-dimensional varieties 37B40 Topological entropy
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