Pure states on O_d

We study representations of the Cuntz algebras O their restrictions to the gauge-invariant subalgebra UHFd have an interesting cyclic structure. If S 1 le i are representatives of the Cuntz relations on a Hilbert space H, special attention is given to the subspaces which are invariant under S. The applications include wavelet multiresolutions corresponding to wavelets of compact support (to appear in the later paper [8]), and finitely correlated states on one-dimensional quantum spin chains.