Title

Iterated Function Systems with Overlaps and Self-Similar Measures

Document Type

Article

Publication Date

2-2001

Publication Title

Journal of London Mathematical Society

DOI

10.1112/S0024610700001654

ISSN

1469-7750

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.

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