Let f:X⇢X be a birational transformation of a
projective manifold X whose Kodaira dimension κ(X) is non-negative. We
show that, if there exist a meromorphic fibration π:X⇢B and a pseudo-automorphism fB:B⇢B which preserves a
big line bundle L∈Pic(B) and such that fB∘π=π∘f, then
fB has finite order.
As a corollary we show that, for projective irreducible symplectic manifolds
of type K3[n] or generalized Kummer, the first dynamical degree
characterizes the birational transformations admitting a Zariski-dense orbit