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

Let $ N \in \mathbb{N} $, $ N \geq 2 $, be given. Motivated by wavelet analysis, we consider a class of normal representations of the $ C^* $-algebra $ \mathfrak{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 \in L^{1}(\mathbb{T}) $, $ h \geq 0 $, to $ R(h)=h $ where $ R $ is a certain transfer operator (positivity-preserving) which was studied previously by D. Ruelle. The representations of $ \mathfrak{A}_{N} $ may also be viewed as representations of a certain (discrete) $ N $-adic $ ax+b $ group which was considered recently by J.-B. Bost and A. Connes.