Affine systems: Asymptotics at infinity for fractal measures

作者：Jorgensen Palle E T^*; Kornelson Keri A; Shuman Karen L

来源：Acta Applicandae Mathematicae, 2007, 98(3): 181-222.

DOI：10.1007/s10440-007-9156-4

• Fourier analysis
• iterated function system
• overlap
• fractal
• measures in product spaces

We study measures on R-d supercript stop 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 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 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 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.

• 出版日期2007-9