Journal article
Affine Systems: Asymptotics at Infinity for Fractal Measures
Acta Applicandae Mathematicae, Vol.98(3), pp.181-222
09/2007
DOI: 10.1007/s10440-007-9156-4
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.
Details
- Title: Subtitle
- Affine Systems: Asymptotics at Infinity for Fractal Measures
- Creators
- Palle Jorgensen - Department of Mathematics The University of Iowa Iowa City IA 52242-1419 USAKeri Kornelson - Department of Mathematics and Statistics Grinnell College Grinnell IA 50112-1690 USAKaren Shuman - Department of Mathematics and Statistics Grinnell College Grinnell IA 50112-1690 USA
- Resource Type
- Journal article
- Publication Details
- Acta Applicandae Mathematicae, Vol.98(3), pp.181-222
- DOI
- 10.1007/s10440-007-9156-4
- ISSN
- 0167-8019
- eISSN
- 1572-9036
- Publisher
- Springer Netherlands; Dordrecht
- Language
- English
- Date published
- 09/2007
- Academic Unit
- Mathematics
- Record Identifier
- 9983985916702771
Metrics
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