### 1.3 Outline, the expanding direction

The asymptotic behavior in the expanding direction of polarized Gowdy. Only in the case of
T^{3}-topology is there an expanding direction. Consequently, it is only necessary to discuss the general
T^{3}-case. However, there are some results of interest, which are only known in the polarized case.
Consequently, we devote Section 12 to a discussion of it. It is of particular interest to note that the spatial
variation of solutions dies out in the sense that the difference between the solution and its average converges
to zero. On the other hand, with respect to other measures, the solution does not tend to spatial
homogeneity.
The asymptotic behavior in the expanding direction of general T^{3}-Gowdy. In the general
case, less detailed information is available. However, a clear picture of the asymptotics exists and is
described in Section 13. The first step of the analysis consists of proving that a naturally defined energy
converges to zero at a specific rate. This leads to the conclusion that the distance from the
solution to its average converges to zero. In order to analyze the asymptotics of solutions, it is
convenient to note that there are conserved quantities. When viewed in the right way, these
conserved quantities can be reinterpreted as ODEs for the averages, and this leads to detailed
information concerning their asymptotics. Finally, results concerning the decay of the sup norm
of derivatives is derived. Such estimates are useful in order to prove future causal geodesic
completeness.