Log Calabi-Yau structure of algebaic varieties admitting a polarized endomorphism

Let $X$ be a normal projective variety admitting a polarized endomorphism $f$, i.e., $f^*H\sim qH$ for some ample divisor $H$ and integer $q>1$. Then Broustet and Gongyo proposed the conjecture that $X$ is of Calabi-Yau type (CY for short), i.e., $(X,\Delta)$ is lc for some effective $\mathbb{Q}$-divisor $\Delta$ and $K_X+\Delta\sim_{\mathbb{Q}}0$. We prove the conjecture when $X$ is a Gorenstein terminal 3-fold, extending the result of Sheng Meng for smooth threefolds. We then study the singularity type and CY property for $(X,\Delta+\frac{R_{\Delta}}{q-1})$ when $(X,\Delta)$ is an $f$-pair, i.e., $K_X+\Delta=f^*(K_X+\Delta)+R_\Delta$ with $\Delta, R_{\Delta}$ being effective. In particular, we show: (1) $K_X + \frac{R_f}{q-1}$ is $\mathbb{Q}$-Cartier and numerically trivial when $X$ is a $\mathbb{Q}$-factorial (or of klt type) $3$-fold; (2) $(X, \frac{R_{f}}{q-1})$ is log Calabi-Yau when $X$ is a surface with the Picard number $\rho(X)>1$ or $f^{-s}(P)=P$ for some prime divisor $P$ and $s>0$.