Automorphisms of $K3$ surfaces, signatures, and isometries of lattices

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Let \alpha be a Salem number of degree d with 4 \leqslant d \leqslant 18 . We show that if d \equiv 0, 4 \ {\rm or}\ 6 \allowbreak{\rm (mod \ 8)} , then \alpha is the dynamical degree of an automorphism of a complex (non-projective) K3 surface. We define a notion of signature of an automorphism, and use it to give a criterion for Salem numbers of degree 10 and 18 to be realized as the dynamical degree of such an automorphism. The first part of the paper contains results on isometries of lattices.