Iterated function systems, moments, and transformations of infinite matrices

TL;DR

This paper explores the relationship between iterated function systems, their equilibrium measure moments, and infinite matrices, establishing a new operator-theoretic framework for understanding geometric transformations.

Contribution

It introduces a novel correspondence between IFS moments and operators in Hilbert space, linking geometric transformations to operator transformations.

Findings

Established a new proof for the existence of measures with specified moments.

Derived conditions for moment matrices to satisfy intertwining relations in IFSs.

Analyzed spectral properties of moment matrices using the Hilbert matrix as a key example.

Abstract

We study the moments of equilibrium measures for iterated function systems (IFSs) and draw connections to operator theory. Our main object of study is the infinite matrix which encodes all the moment data of a Borel measure on R^d or C. To encode the salient features of a given IFS into precise moment data, we establish an interdependence between IFS equilibrium measures, the encoding of the sequence of moments of these measures into operators, and a new correspondence between the IFS moments and this family of operators in Hilbert space. For a given IFS, our aim is to establish a functorial correspondence in such a way that the geometric transformations of the IFS turn into transformations of moment matrices, or rather transformations of the operators that are associated with them. We first examine the classical existence problem for moments, culminating in a new proof of the existence of a Borel measure on R or C with a specified list of moments. Next, we consider moment problems associated with affine and non-affine IFSs. Our main goal is to determine conditions under which an intertwining relation is satisfied by the moment matrix of an equilibrium measure of an IFS. Finally, using the famous Hilbert matrix as our prototypical example, we study boundedness and spectral properties of moment matrices viewed as Kato-Friedrichs operators on weighted l^2 spaces.