DIMENSION PROPERTIES OF RANDOM FRACTALS WITH OVERLAPS

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 Falconer, Graf, Mauldin and Williams 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).