  We prove that the collection $\mathcal M_{-\infty}$ of backward bounded
solutions for a semilinear evolution equation is the graph of an upper
hemicontinuous set-valued function from the low Fourier modes to the higher
Fourier modes, which is invariant and contains the global attractor. We also
show that there exists a limit $\mathcal M_{\infty}$ of finite dimensional
Lipschitz manifolds $\mathcal M_t$ generated by the time $t$-maps ($t>0$) from
the flat manifold $\mathcal M_0$ with the Hausdorff distance and we find
$\mathcal M_{\infty} \subset \mathcal M_{-\infty}$. No spectral gap conditions
are assumed.
