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Dimension properties of random fractals with overlaps. (English) Zbl 1074.60049
Summary: We consider random fractals generated by random recursive constructions with overlaps. Our construction allows some overlaps among sets in the same generation. We introduce a certain “limited overlaps condition”. Under this condition, we prove that the Hausdorff dimension of the generated fractal satisfies the expectation equation (upon non-extinction), which was studied previously by K. J. Falconer [Math. Proc. Camb. Philos. Soc. 100, 559–582 (1986; Zbl 0623.60020)], S. Graf [Probab. Theory Relat. Fields 74, 357–392 (1987; Zbl 0591.60005)] and R. D. Mauldin and S. C. Williams [Trans. Am. Math. Soc. 295, 325–346 (1986; Zbl 0625.54047)] under open set condition. We also prove that the generated fractal is regular in the sense that its Hausdorff and upper box dimension are equal to a non-random constant (this result holds without assumption of limited overlaps condition).
60G17 Sample path properties
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