Some constraints on positive entropy automorphisms of smooth threefolds

Suppose that X is a smooth, projective threefold over C and that Φ W X ! 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 in nitely many KX-negative extremal rays on NE.X/.