Polarized endomorphisms of log Calabi-Yau pairs

Let $(X,\mathrm{\Delta })$ be a dlt log Calabi-Yau pair admitting a polarized endomorphism. We show that $(X,\mathrm{\Delta })$ 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 $(X,\mathrm{\Delta })$ even if $X$ is a smooth variety. Given a klt type variety $X$ and a log Calabi-Yau pair $(X,\mathrm{\Delta })$ admitting a polarized endomorphism, we show that a suitable birational modification of $(X,\mathrm{\Delta })$ is a finite quotient of a toric log Calabi-Yau fibration over an abelian variety.