TY - JOUR
T1 - Iterated Function Systems with Overlaps and Self-Similar Measures
AU - Lau, Ka-Sing
AU - Ngai, Sze-Man
AU - Rao, Hui
PY - 2001/2/1
Y1 - 2001/2/1
N2 - 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.
AB - 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.
KW - Iterated Function Systems
KW - Overlaps
KW - Self-similar Measures
UR - https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/607
UR - https://doi.org/10.1112/S0024610700001654
U2 - 10.1112/S0024610700001654
DO - 10.1112/S0024610700001654
M3 - Article
VL - 63
JO - Journal of London Mathematical Society
JF - Journal of London Mathematical Society
ER -