Polarized endomorphisms of log Calabi-Yau pairs

Let (X, ) be a dlt log Calabi-Yau pair admitting a polarized endomorphism. We show that (X, ) is a finite quotient of a toric log Calabi-Yau fibration over an abelian variety. We provide an example which shows that the previous statement does not hold if we drop the dlt condition of (X, ) even if X is a smooth variety. Given a klt type variety X and a log Calabi-Yau pair (X, ) admitting a polarized endomorphism, we show that a suitable birational modification of (X, ) is a finite quotient of a toric log Calabi-Yau fibration over an abelian variety.