  In the renormalisation analysis of critical phenomena in quasi-periodic
systems, a fundamental role is often played by fixed points of functional
recurrences of the form \begin{equation*} f_{n}(x) = \sum_{i=1}^\ell a_i(x)
f_{n_i} (\alpha_i(x)) \,, \end{equation*} where the $\alpha_i$, $a_i$ are known
functions and the $n_i$ are given and satisfy $n-2 \le n_i \le n-1 $. We
develop a general theory of fixed points of ``Composition Sum Operators''
derived from such recurrences, and apply it to test for fixed points in key
classes of complex analytic functions with singularities. Finally we
demonstrate the construction of the full space of fixed points of one important
class, for the much studied operator \begin{equation*} Mf(x) = f(-\omega x) +
f(\omega^2 x + \omega)\,, \quad \omega = (\sqrt{5}-1)/2\,. \end{equation*} The
construction reveals previously unknown solutions.
