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Let $X$ be a projective variety and $\sigma$ a wild automorphism on $X$,i.e., whenever $\sigma(Z) = Z$ for a non-empty Zariski-closed subset $Z$ of$X$, we have $Z = X$. Then $X$ is conjectured to be an abelian variety with$\sigma$ of zero entropy (and proved to be so when ${\rm dim} \, X \le 2$) byZ. Reichstein, D. Rogalski and J. J. Zhang in their study of projectivelysimple rings. This conjecture has been generally open for more than a decade.In this note, we confirm this original conjecture when ${\rm dim} \, X \le 3$and $X$ is not a Calabi-Yau threefold, and also show that $\sigma$ is of zeroentropy when ${\rm dim} \, X \le 4$ and the Kodaira dimension $\kappa(X) \ge0$.

Wild automorphisms of projective varieties, the maps which have no invariant proper subsets