A criterion for the primitivity of a birational automorphism of a Calabi-Yau manifold and an application

We shall give a sufficient condition on the primitivity of a birational automrphism of a Calabi-Yau manifold in purely algebro geometric terms. As an application, we shall give an explicit construction of Calabi-Yau manifolds of Picard number $2$ of any dimension $\ge 3$, with primitive birational automrphisms of first dynamical degree $>1$.