Special valuations and automorphisms of affine log Calabi--Yau varieties

Let $U$ be an affine log Calabi--Yau variety. Although special and finitely generated valuations are defined using a log CY-Fano compactification of $U$, we prove that the corresponding skeleta in the dual complex are independent of this choice, and hence invariant under $\mathrm{Aut}(U)$. We also s