Dimensions of random recursive sets

We prove a theorem that generalizes the equality among the packing, Hausdorff, and upper and lower Minkowski dimensions for a general class of random recursive constructions, and apply it to constructions with finite memory. Then we prove an upper bound on the packing dimension of certain random distribution functions on [0, 1]. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 328, 2005, pp. 20–26.     

Packing Measure and Dimension of Random Fractals

We consider random fractals generated by random recursive constructions. We prove that the box-counting and packing dimensions of these random fractals, K, equals , their almost sure Hausdorff dimension. We show that some almost deterministic conditions known to ensure that the Hausdorff measure satisfies also imply that the packing measure satisfies 0< . When these conditions are not satisfied, it is known . Correspondingly, we show that in this case , provided a random strong open set condition is satisfied. We also find gauge functions (t) so that the -packing measure is finite.     

Exact packing dimension in random recursive constructions

We explore the exact packing dimension of certain random recursive constructions. In case of polynomial decay at 0 of the distribution function of random variable X, associated with the construction, we prove that it does not exist, and in case of exponential decay it is t|log|logt||, where is the fractal dimension of the limit set and 1/ is the rate of exponential decay.Research supported by the Department of Mathematics and Statistics (Mathematics) at University of Jyväskylä.Mathematics Subject Classification (2000):Primary 28A78, 28A80; Secondary 60D05, 60J80