  In this paper, we introduce a property of topological dynamical systems that
we call finite dynamical complexity. For systems with this property, one can in
principle compute the $K$-theory of the associated crossed product
$C^*$-algebra by splitting it up into simpler pieces and using the methods of
controlled $K$-theory. The main part of the paper illustrates this idea by
giving a new proof of the Baum-Connes conjecture for actions with finite
dynamical complexity.
  We have tried to keep the paper as self-contained as possible: we hope the
main part will be accessible to someone with the equivalent of a first course
in operator $K$-theory. In particular, we do not assume prior knowledge of
controlled $K$-theory, and use a new and concrete model for the Baum-Connes
conjecture with coefficients that requires no bivariant $K$-theory to set up.
