Summary
The self-similarity of sets (measures) is often defined in a constructive way. In the present paper it will be shown that the random recursive construction model of Falconer, Graf and Mauldin/Williams for (statistically) self-similar sets may be generalized to the noncompact case. We define a sequence of random finite measures, which converges almost surely to a self-similar random limit measure. Under certain conditions on the generating Lipschitz maps we determine the carrying dimension of the limit measure.
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Arbeiter, M. Random recursive construction of self-similar fractal measures. The noncompact case. Probab. Th. Rel. Fields 88, 497–520 (1991). https://doi.org/10.1007/BF01192554
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DOI: https://doi.org/10.1007/BF01192554