Some constraints on positive entropy automorphisms of smooth threefolds

Suppose that $X$ is a smooth, projective threefold over $\mathbb C$ and that $ϕ: X \to X$ is an automorphism of positive entropy. We show that one of the following must hold, after replacing $ϕ$ by an iterate: i) the canonical class of $X$ is numerically trivial; ii) $ϕ$ is imprimitive; iii) $ϕ$ is not dynamically minimal. As a consequence, we show that if a smooth threefold $M$ does not admit a primitive automorphism of positive entropy, then no variety constructed by a sequence of smooth blow-ups of $M$ can admit a primitive automorphism of positive entropy. In explaining why the method does not apply to threefolds with terminal singularities, we exhibit a non-uniruled, terminal threefold $X$ with infinitely many $K_X$-negative extremal rays on $NE(X)$.

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