Ruelle operators: Functions which are harmonic with respect to a transfer operator

Let N is an element of W, N greater than or equal to 2, be given. Motivated by wavelet analysis, we consider a class of normal representations of the C*-algebra A(N) on two unitary generators U, V subject to the relation UVU-1 = V-N The representations are in one-to-one correspondence with solutions h is an element of L-1 (T), h greater than or equal to 0, to R(h) = h where R is a certain transfer operator (positivity-preserving) which was studied previo usly by D. Ruelle. The representations of A(N) may also be viewed as repres entations of a certain (discrete) N-adic ax + b group which was considered recently by J.-B. Best and A. Connes.