Fixed points of Composition Sum Operators

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.