Affine Systems: Asymptotics at Infinity for Fractal Measures

Abstract

We study measures on 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 μ induced by a finite family of affine mappings in 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 μ at hand (frequency domain), we identify asymptotic laws, spectral types, dichotomy, and chaos laws. In particular we show that the cases when μ 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 μ for paths at infinity in R d . We show how properties of μ depend on perturbations of the initial data, e.g., variations in a prescribed finite set of affine mappings in 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.

Families of Spectral Sets for Bernoulli Convolutions

“…This general framework includes subjects such as universal tiling sets and the dual spectral set conjecture (see [11,22,25,27,35]). Starting with [16], questions about Fourier duality have received considerable attention with respect to pure harmonic analysis [5,8,9,24,29,30,33,34] and with respect to applications such as wavelets, sampling, algorithms, martingales, and substitution-dynamical systems [2][3][4]26].…”

Non-spectral problem for a class of planar self-affine measures

“…The self-affine measure μ M,D and its Fourier transformμ M,D given by (2.2) play an important role in analysis and geometry. Previous research on such measure and its Fourier transform revealed some surprising connections with a number of areas in mathematics, such as harmonic analysis, dynamical systems, number theory, and others, see [8,9,20,30,31] and references cited therein. Here we are interested in the zero set Z(μ M,D ) ofμ M,D which is highly important to the spectral and non-spectral problems on the self-affine measures.…”

Continuous dependence on parameters of certain self-affine measures, and their singularity

“…The previous research on such a measure and its Fourier transform revealed some surprising connections with a number of areas in mathematics, such as harmonic analysis, number theory, dynamical systems, and others (see, e.g. [5], [9], [13], [15]). The previous studies have also left some well-known open problems, such as the nature of the Bernoulli convolutions (cf.…”