  We study representations of the Cuntz algebras O_d and their associated
decompositions. In the case that these representations are irreducible, their
restrictions to the gauge-invariant subalgebra UHF_d have an interesting cyclic
structure. If S_i, 1 \leq i \leq d, are representatives of the Cuntz relations
on a Hilbert space H, special attention is given to the subspaces which are
invariant under S_i^*. The applications include wavelet multiresolutions
corresponding to wavelets of compact support (to appear in the later paper
\cite{BEJ97}), and finitely correlated states on one-dimensional quantum spin
chains.
