Affine systems: asymptotics at infinity for fractal measures

TL;DR

This paper investigates the asymptotic behavior of measures generated by affine and recursive systems, revealing a spectrum of properties from fractal to chaotic, through Fourier analysis and perturbation studies.

Contribution

It introduces a systematic Fourier transform approach to analyze the asymptotics and spectral types of measures from affine and recursive systems, including overlap cases and complex dynamics.

Findings

Identifies asymptotic laws and spectral types of measures.

Shows a gradation from fractal to chaotic measures based on initial data.

Demonstrates sensitivity of measures to perturbations in system parameters.

Abstract

We study measures on $\mathbb{R}^d$ which are induced by a class of infinite and recursive iterations in symbolic dynamics. Beginning with a finite set of data, we analyze prescribed recursive iteration systems, each involving subdivisions. The construction includes measures arising from affine and contractive iterated function systems with and without overlap (IFSs), i.e., limit measures $\mu$ induced by a finite family of affine mappings in $\mathbb{R}^d$ (the focus of our paper), as well as equilibrium measures in complex dynamics. By a systematic analysis of the Fourier transform of the measure $\mu$ at hand (frequency domain), we identify asymptotic laws, spectral types, dichotomy, and chaos laws. In particular we show that the cases when $\mu$ is singular carry a gradation, ranging from Cantor-like fractal measures to measures exhibiting chaos, i.e., a situation when small changes in the initial data produce large fluctuations in the outcome, or rather, the iteration limit (in this case the measures). Our method depends on asymptotic estimates on the Fourier transform of $\mu$ for paths at infinity in $\mathbb{R}^d$. We show how properties of $\mu$ depend on perturbations of the initial data, e.g., variations in a prescribed finite set of affine mappings in $\mathbb{R}^d$, in parameters of a rational function in one complex variable (Julia sets and equilibrium measures), or in the entries of a given infinite positive definite matrix.