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1995 Densities of self-similar measures on the line
Robert S. Strichartz, Arthur Taylor, Tong Zhang
Experiment. Math. 4(2): 101-128 (1995).

Abstract

We describe algorithms to compute self-similar measures associated to iterated function systems (i.f.s.) on an interval, and more general self-replicating measures that include Hausdorff measure on the attractor of a nonlinear i.f.s. We discuss a variety of error measurements for these algorithms. We then use the algorithms to study density properties of these measures experimentally. By density we mean the behavior of the ratio $\mu(B_r(x))/(2r)^\alpha$ as $r \rightarrow 0$, were $\alpha$ is an appropriate dimension. It is well-known that a limit usually does not exist. We have found an intriguing structure associated to these ratios that we call density diagrams. We also use density computations to approximate the exact Hausdorff measure of the attractor of an i.f.s.

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Robert S. Strichartz. Arthur Taylor. Tong Zhang. "Densities of self-similar measures on the line." Experiment. Math. 4 (2) 101 - 128, 1995.

Information

Published: 1995
First available in Project Euclid: 17 March 2003

zbMATH: 0860.28005
MathSciNet: MR1377413

Subjects:
Primary: 28A78
Secondary: 26A30 , 58F11

Rights: Copyright © 1995 A K Peters, Ltd.

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Vol.4 • No. 2 • 1995
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