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Let $X$ be an abelian variety over an algebraically closed field $\mathbf{k}$and $f$ a quasi-unipotent automorphism of $X$. When $\mathbf{k}$ is the fieldof complex numbers, Lin, Oguiso, and D.-Q. Zhang provide an explicit formulafor the polynomial volume growth of (or equivalently, for the Gelfand--Kirillovdimension of the twisted homogeneous coordinate ring associated with) the pair$(X, f)$, by an analytic argument. We give an algebraic proof of this formulathat works in arbitrary characteristic.  In the course of the proof, we obtain: (1) a new description of the action ofendomorphisms on the $\ell$-adic Tate spaces, in comparison with recent resultsof Zarhin and Poonen--Rybakov; (2) a partial converse to a result ofReichstein, Rogalski, and J.J. Zhang on quasi-unipotency of endomorphisms andtheir pullback action on the rational N\'eron--Severi space$\mathsf{N}^1(X)_{\mathbf{Q}}$ of $\mathbf{Q}$-divisors modulo numericalequivalence; (3) the maximum size of Jordan blocks of (the Jordan canonicalform of) $f^*|_{\mathsf{N}^1(X)_{\mathbf{Q}}}$ in terms of the action of $f$ onthe Tate space $V_\ell(X)$.

Polynomial volume growth of quasi-unipotent automorphisms of abelian varieties (with an appendix in collaboration with Chen Jiang)