  In this paper we study systems of $N$ uniformly expanding coupled maps when
$N$ is finite but large. We introduce self-consistent transfer operators that
approximate the evolution of measures under the dynamics, and quantify this
approximation explicitly with respect to $N$. Using this result, we prove that
uniformly expanding coupled maps satisfy propagation of chaos when
$N\rightarrow \infty$, and characterize the absolutely continuous invariant
measures for the finite dimensional system. The main working assumption is that
the expansion is not too small and the strength of the interactions is not too
large, although both can be of order one. In contrast with previous approaches,
we do not require the coupled maps and the interactions to be identical. The
technical advances that allow us to describe the system are: the introduction
of a framework to study the evolution of conditional measures along some
non-invariant foliations where the dependence of all estimates on the dimension
is explicit; and the characterization of an invariant class of measures close
to products that satisfy exponential concentration inequalities.
