  Topological dynamics of cellular automata (CA), inherited from classical
dynamical systems theory, has been essentially studied in dimension 1. This
paper focuses on higher dimensional CA and aims at showing that the situation
is different and more complex starting from dimension 2. The main results are
the existence of non sensitive CA without equicontinuous points, the
non-recursivity of sensitivity constants, the existence of CA having only
non-recursive equicontinuous points and the existence of CA having only
countably many equicontinuous points. They all show a difference between
dimension 1 and higher dimensions. Thanks to these new constructions, we also
extend undecidability results concerning topological classification previously
obtained in the 1D case. Finally, we show that the set of sensitive CA is only
Pi_2 in dimension 1, but becomes Sigma_3-hard for dimension 3.
