  To a representation of $\O_N$ (the Cuntz algebra with $N$ generators) we
associate a projection valued measure and we study the case when this measure
has atoms. The main technical tool are the spaces invariant for all the
operators $S_i^*$. We classify the purely atomic representations and find when
such representations are permutative. Applications include: wavelet
representations, representations generated by finitely correlated states,
representation associated to Hadamard triples (or fractal spectral measures)
and representations associated to generalized Walsh bases.
