Let $Ninmathbb{N}$, $Ngeq2$, 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 $hin L^{1}left(mathbb{T}right)$, $hgeq0$, to $Rleft(hright)=h$ 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.