Self-similar measure

Summary

A self-similar measure is a probability measure on $\R^n$ that is a weighted average of its own images under finitely many contracting similarities. For given similarities and weights there is exactly one such measure, by a theorem of Hutchinson. Its support is the self-similar fractal, or attractor, of the similarities. Self-similar measures are the basic examples of fractal measures, and their dimensions and Fourier decay are studied in detail.