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We show that there exists an automorphism of a projective K3 surface with Picard number 2 such that the trace of its action on the Picard lattice is 3. Together with a result of K. Hashimoto, J. Keum and K. Lee, we determine the set of dynamical degrees of automorphisms of projective K3 surfaces with Picard number 2.
Yuta Takada (Thu,) studied this question.