Abstract. Let N ∈ N, N ≥ 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 relationThe representations are in one-to-one correspondence with solutions h ∈ L 1 (T), In multiresolution wavelet theory, there is a fundamental interplay and interconnection between the following two operators: M and R, where M is the cascade refinement operator and R is the corresponding transfer operator. We also denote the second of these, R, the Ruelle operator, because of its close connection to an operator that David Ruelle used first in his study of phase transitions in quantum statistical mechanics lattice models; see [Rue68], [May80],and [Rue78a]. We first recall the two operators here in a simple wavelet context, but the scope will be 1991 Mathematics Subject Classification. Primary 46L60, 47D25, 42A16, 43A65; Secondary 46L45, 42A65, 41A15.