  Motivated by wavelet analysis, we prove that there is a one-to-one
correspondence between the following data: Solutions to $R(h)=h$ where $R$ is a
certain non-positive Ruelle transfer operator; Operators that intertwine a
certain class of representations of the $C^*$-algebra $\mathfrak{A}_N$ on two
unitary generators $U$, $V$ subject to the relation $$UVU^{-1}=V^N$$ This
correspondence enables us to give a criterion for the biorthogonality of a pair
of scaling functions and calculate all solutions of the equation $R(h)=h$ in
some concrete cases.
