Palle E. T Jorgensen 
Ruelle Operators [PDF ebook] 
Functions which Are Harmonic with Respect to a Transfer Operator

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.