Ruelle Operators

Let $N\in\mathbb{N}$, $N\geq2$, 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 $UVU^{-1}=V^{N}$. The representations are in one-to-one correspondence with solutions $h\in L^{1}\left(\mathbb{T}\right)$, $h\geq0$, to $R\left(h\right)=h$ 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.