Acta Applicandae Mathematicae

, Volume 98, Issue 3, pp 181–222

Affine Systems: Asymptotics at Infinity for Fractal Measures

  • Palle E. T. Jorgensen
  • Keri A. Kornelson
  • Karen L. Shuman
Article

DOI: 10.1007/s10440-007-9156-4

Cite this article as:
Jorgensen, P.E.T., Kornelson, K.A. & Shuman, K.L. Acta Appl Math (2007) 98: 181. doi:10.1007/s10440-007-9156-4

Abstract

We study measures on ℝd 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 ℝd (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 ℝd. 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 ℝd, 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.

Keywords

Fourier analysisIterated function systemOverlapFractalMeasures in product spaces

Mathematics Subject Classification (2000)

11K5528A3542A55

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  • Palle E. T. Jorgensen
    • 1
  • Keri A. Kornelson
    • 2
  • Karen L. Shuman
    • 2
  1. 1.Department of MathematicsThe University of IowaIowa CityUSA
  2. 2.Department of Mathematics and StatisticsGrinnell CollegeGrinnellUSA