  We develop a purely categorical theory of action filtrations and their
associated growth invariants. When specialized to categories of geometric
interest, such as the wrapped Fukaya category of a Weinstein manifold, and the
bounded derived category of coherent sheaves on a smooth algebraic variety, our
categorical action filtrations essentially recover previously studied
filtrations of geometric origin.
  Our approach is built around the notion of a smooth categorical
compactification. We prove that a smooth categorical compactification induces
well-defined growth invariants, which are moreover preserved under zig-zags of
such compactifications. The technical heart of the paper is a method for
computing these growth invariants in terms of the growth of certain colimits of
(bi)modules. In practice, such colimits arise in both geometric settings of
interest.
  The main applications are: (1) A "quantitative" refinement of homological
mirror symmetry, which relates the growth of the Reeb-length filtration on the
symplectic geometry side with the growth of filtrations on the algebraic
geometry side defined by the order of pole at infinity (often these can be
expressed in terms of the dimension of the support of sheaves). (2) A proof
that the Reeb-length growth of symplectic cohomology and wrapped Floer
cohomology on a Weinstein manifold are at most exponential. (3) Lower bounds
for the entropy and polynomial entropy of certain natural endofunctors acting
on Fukaya categories.
