An application of $p$-adic integration to the dynamics of a birational transformation preserving a fibration

Abstract

Let $f:X⇢X$ be a birational transformation of a projective manifold $X$ whose Kodaira dimension $\kappa (X)$ is non-negative. We show that, if there exist a meromorphic fibration $\pi :X⇢B$ and a pseudo-automorphism $f_B:B⇢B$ which preserves a big line bundle $L\in Pic(B)$ and such that $f_B\circ \pi =\pi \circ f$, then $f_B$ has finite order. As a corollary we show that, for projective irreducible symplectic manifolds of type $K{3}^{[n]}$ or generalized Kummer, the first dynamical degree characterizes the birational transformations admitting a Zariski-dense orbit.