For every q-polarized endomorphism of P^2, some iterate (P^2, R_{f^s}/(q^s-1)) is log canonical, settling Gongyo's conjecture for smooth projective surfaces.
Blum,Existence of valuations with smallest normalized volume, Compos
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Log Calabi--Yau structure for endomorphisms on $\mathbf{P}^n$
For every q-polarized endomorphism of P^2, some iterate (P^2, R_{f^s}/(q^s-1)) is log canonical, settling Gongyo's conjecture for smooth projective surfaces.