terms mentioned above uniquely determine the variables, and to write down a [*Fuchsian*]{} system for the decaying quantities which is equivalent to the Einstein-matter evolution system. As the use of Fuchsian systems is central to our work let us briefly recall what a Fuchsian system is and how such a system is related to familiar iterative methods. For a more detailed introduction to Fuchsian techniques see [@AR; @KR; @R00a; @Rendall:2001ai] and references therein. Note that we shall everywhere restrict ourselves to the analytic case. We expect that our results extend to the $C^{\infty}$ case, but it is a non-trivial task to prove that they do.

The general form of a Fuchsian system for a vector-valued unknown function $u$ is $$\label{fuchs0}
t \, \partial_t u + {\cal A}(x) \, u = f(t,x,u,u_x),$$ where the matrix ${\cal A}(x)$ is required to satisfy some positivity condition (see below), while the “source term” $f$ on the right hand side is required to be “regular.” (See [@AR] for precise criteria allowing one to check when the positivity assumption on ${\cal A}(x)$ is satisfied and when $f$ is regular.) A key point is that $f$ is required to be bounded by terms of order $O(t^{\delta})$ (with 