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ITERATED FUNCTION SYSTEMS WITH OVERLAPS AND SELF-SIMILAR MEASURES

Published online by Cambridge University Press:  19 March 2001

KA-SING LAU ,
SZE-MAN NGAI  and
HUI RAO
KA-SING LAU
Affiliation:
Department of Mathematics, Chinese University of Hong Kong, Hong Kong; kslau@math.cuhk.edu.hk
SZE-MAN NGAI
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA Current address of S. M. Ngai: Department of Mathematics, Georgia Southern University, Statesboro, GA 30460, USA; ngai@gsu.cs.gasou.edu
HUI RAO
Affiliation:
Department of Mathematics, Wuhan University, Wuhan 430072, China; raohui@nlsc.whu.edu.cn
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Abstract

The paper considers the iterated function systems of similitudes which satisfy a separation condition weaker than the open set condition, in that it allows overlaps in the iteration. Such systems include the well-known Bernoulli convolutions associated with the PV numbers, and the contractive similitudes associated with integral matrices. The latter appears frequently in wavelet analysis and the theory of tilings. One of the basic questions is studied: the absolute continuity and singularity of the self-similar measures generated by such systems. Various conditions to determine the dichotomy are given.

Type
Research Article
Information
Journal of the London Mathematical Society , Volume 63 , Issue 1 , February 2001 , pp. 99 - 116
Copyright
The London Mathematical Society 2001

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