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

Twisted homogeneous coordinate rings are natural invariants associated to a projective variety X with an automorphism f. We study the Gelfand-Kirillov dimensions of these noncommutative algebras from the perspective of complex dynamics, by noticing that when X is a smooth complex projective variety, they essentially coincide with the polynomial logarithmic volume growth Plov(f) of (X,f). We formulate some basic dynamical properties about these invariants and study explicit examples. Our main results are new uniform upper bounds of these invariants, in terms of the dimension of the variety (as well as other refinements).