Harmonic analysis and dynamics for affine iterated function systems

TL;DR

This paper develops a harmonic analysis framework for affine iterated function systems in multi-dimensional space, introducing duality concepts and identities for Fourier transforms of measures related to Bernoulli convolutions.

Contribution

It introduces a novel harmonic analysis approach for affine IFSs, including duality notions and Fourier transform identities, advancing understanding of these systems.

Findings

Established a duality notion for affine and contractive IFSs

Derived identities for Fourier transforms of measures

Extended harmonic analysis techniques to multi-dimensional affine systems

Abstract

We introduce a harmonic analysis for a class of affine iteration models in $\br^d$. Using Hilbert-space geometry, we develop a new duality notion for affine and contractive iterated function systems (IFSs) and we construct some identities for the Fourier transform of the measure corresponding to infinite Bernoulli convolutions.