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

Let $N\in \mathbb{N}$, $N\ge 2$, be given. Motivated by wavelet analysis, we consider a class of normal representations of the $C^{\ast }$-algebra ${\mathfrak{A}}_N$ on two unitary generators $U$, $V$ subject to the relation $$UV{U}^{-1}=V^N\mathrm{.}$$ The representations are in one-to-one correspondence with solutions $h\in L^1(\mathbb{T})$, $h\ge 0$, 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.