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Let f be a zero entropy automorphism of a compact K\"ahler manifold X. Westudy the polynomial log-volume growth Plov(f) of f in light of the dynamicalfiltrations introduced in our previous work with T.-C. Dinh. We obtain newupper bounds and lower bounds of Plov(f). As a corollary, we completelydetermine Plov(f) when dim X = 3, extending a result of Artin--Van den Berghfor surfaces. When X is projective, Plov(f) + 1 coincides with theGelfand--Kirillov dimensions GKdim(X,f) of the twisted homogeneous coordinaterings associated to (X,f). Reformulating these results for GKdim(X,f), weimprove Keeler's bounds of GKdim(X,f) and provide effective upper bounds ofGKdim(X,f) which only depend on dim X.

Polynomial log-volume growth in slow dynamics and the GK-dimensions of twisted homogeneous coordinate rings