Iterated function systems and permutation representations of the Cuntz algebra

We study a class of representations of the Cuntz algebras ON, N = 2, 3,..., acting on L-2(T) where T = R/2 pi Z. The representations arise in wavelet theory, but are of independent interest. We find and describe the decomposi tion into irreducibles, and show how the O-N-irreducibles decompose when re stricted to the subalgebra UHFN subset of O-N of gauge-invariant elements; and we show that the whole structure is accounted for by arithmetic and com binatorial properties of the integers Z. We have general. results on a clas s of representations of O-N on Hilbert space H such that the generators Si as operators permute the elements in some orthonormal basis for H. We then use this to extend our results from L-2(T) to L-2 (T-d), d > 1; even to L-2 (T) where T is some fractal version of the torus which carries more of the algebraic information encoded in our representations.