  We report the results of the first study of the multi-stream environment of
dark matter haloes in cosmological N-body simulations in the LCDM cosmology.
The full dynamical state of dark matter can be described as a three-dimensional
sub-manifold in six dimensional phase space - the dark matter sheet. In our
study we use a Lagrangian sub-manifold x = x(q,t) (where x and q are co-moving
Eulerian and Lagrangian coordinates respectively), which is dynamically
equivalent to the dark matter sheet but is more convenient for numerical
analysis. Our major results can be summarized as follows. At the resolution of
the simulation i.e. without additional smoothing, the cosmic web represents a
hierarchical structure: each halo is embedded in the filamentary framework of
the web predominantly at the filament crossings, and each filament is embedded
in the wall like fabric of the web at the wall crossings. Locally, each halo or
sub-halo is a peak in the number of streams field. The number of streams in the
neighbouring filaments is higher than in the neighbouring walls. The walls are
regions where number of streams is equal to three or a few. Voids are uniquely
defined by the local condition requiring to be a single-stream flow region. The
shells of streams around haloes are quite thin and the closest void region is
typically within one and a half FOF radius from the center of the halo.
