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.