Fixed points of composition sum operators

Verschueren, Paul and Mestel, Ben D. (2014). Fixed points of composition sum operators. Journal of Difference Equations and Applications, 20(8) pp. 1152–1168.

DOI: https://doi.org/10.1080/10236198.2014.899357

Abstract

In the renormalization analysis of critical phenomena in quasi-periodic systems, a fundamental role is often played by fixed points of functional recurrences of the form

f_n(z) = ^ℓΣ_i=1a_i(z)f_ni(α_i(z)),

where the α_i are affine contractions and each n_i is either n - 1 or n - 2. We develop a general theory of these fixed points by regarding them as fixed points of ‘composition sum operators’, and apply this theory to test for fixed points in classes of complex analytic functions with various key types of singularities. Finally we demonstrate the construction of the full space of fixed points of one important class, arising from the much studied operator M defined by

Mf(z) = f(-ωz) + f(ω²z + ω), ω = (√5 - 1)/2.

The construction reveals previously unknown solutions.

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