Year: 2013
Type: article
Abstract: 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} (α_i(x)) \,, \end{equation*} where the $α_i$, $a_i$ 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(-ωx) + f(ω^2 x + ω)\,, \quad ω= (\sqrt{5}-1)/2\,. \end{equation*} The construction reveals previously unknown solutions.
Authors: Verschueren, Paul, Mestel, Ben D.
Institution: The Open University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2951144134