Quasiperiodic Renormalisation: Quasiperiodic Sums, Products and Composition Sum Operators

Abstract

We study two topics in the renormalisation group theory of quasiperiodic systems, both topics having a number of important applications. Firstly we apply renormalisation techniques to a family of functions we designate quasiperiodic sums and products. These functions link the study of critical phenomena in diverse fields such as the emergence of strange non-chaotic attractors, critical KAM theory, convergence of ergodic averages, and q-series (much used in string theory). In pure mathematics they have also been studied extensively in complex power series analysis, partition theory, and Diophantine approximation. We set these currently dispersed results within a unified framework, and develop a more systematic approach to three key examples. We improve significantly on a series of number-theoretic results obtained by Sierpinsky, Hardy, Hecke, Lang et al; we provide a rigorous proof of some empirically obtained results recently reported by Knill et al (2011-12); and we settle (negatively) an open question of Erdos-Szekeres-Lubinsky dating initially from 1959. Secondly we study the fixed points of quasiperiodic renormalisation. These arise in the study of the Harper equation (almost Mathieu equation), barrier billiards, and a number of other scenarios. We identify the natural setting for their study as a certain class of linear operators (composition sum operators) acting on unbounded (non-Banach) function spaces. We develop the necessary foundations for this theory, and then apply it to construct the space of all fixed points of the golden renormalisation whose singularities are either poles, essential singularities, or logarithmic singularities of a certain simple type. This fixed point space is shown to include previously unknown fixed points.