Hausdorff dimension, Mean quadratic variation of infinite self-similar measures

Under weaker condition than that of Riedi & Mandelbrot, the Hausdorff (and Hausdorff-Besicovitch) dimension of infinite self-similar set K which is the invariant compact set of infinite contractive similarities {S_j(x)} satisfying open set condition is obtained. It is proved (under some additional hypotheses) that the mean quadratic variation of infinite self-similar measure is of asymptotic property.