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