ar X iv : f un ct - a n / 96 12 00 2 v 1 1 7 D ec 1 99 6 ITERATED FUNCTION SYSTEMS AND PERMUTATION REPRESENTATIONS OF THE CUNTZ ALGEBRA

We study a class of representations of the Cuntz algebras O N , N = 2, 3,. .. , acting on L 2 (T) where T = R2πZ. The representations arise in wavelet theory, but are of independent interest. We find and describe the decomposition into irreducibles, and show how the O N-irreducibles decompose when restricted to the subalgebra UHF N ⊂ O N of gauge-invariant elements; and we show that the whole structure is accounted for by arithmetic and combinatorial properties of the integers Z. We have general results on a class of representations of O N on Hilbert space H such that the generators S i 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.