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Iterated Function Systems with Overlaps and Self‐Similar Measures
Author(s) -
Lau KaSing,
Ngai SzeMan,
Rao Hui
Publication year - 2001
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610700001654
Subject(s) - iterated function system , bernoulli's principle , iterated function , mathematics , singularity , function (biology) , pure mathematics , set (abstract data type) , bernoulli distribution , mathematical analysis , computer science , random variable , statistics , attractor , evolutionary biology , engineering , biology , programming language , aerospace engineering
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.