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

Let f be a zero entropy automorphism of a compact Kähler manifold X . We study the polynomial log-volume growth \mathrm{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 \mathrm{Plov}(f) . As a corollary, we completely determine \mathrm{Plov}(f) when \dim X = 3 , extending a result of Artin–Van den Bergh for surfaces. When X is projective, \mathrm{Plov}(f) + 1 coincides with the Gelfand–Kirillov dimensions \mathrm{GKdim}(X,f) of the twisted homogeneous coordinate rings associated to (X,f) . Reformulating these results for \mathrm{GKdim}(X,f) , we improve Keeler’s bounds of \mathrm{GKdim}(X,f) and provide effective upper bounds of \mathrm{GKdim}(X,f) which only depend on \dim X .