Automorphisms of bounded growth

We study birational automorphisms of algebraic varieties of bounded growth, i.e. such that the norms of the inverse images (fⁿ)^* NS(X)→ NS(X) of the powers of the automorphism f(X) are bounded above for n 0. We prove that some power of an infinite order automorphism of a variety X with such property factors either through an infinite order translation on the Albanese variety of X or through an infinite order regular automorphism of Pᵐ for m 1. We deduce from this that if a rationally connected threefold admits an infinite order automorphism whose growth is bounded then the threefold is rational and an iterate of the automorphism is birationally conjugate to a regular automorphism of P³, a generalization of Blanc and Deserti's result.