On multifractal of Cantor dust

Abstract

We consider quasi-self-similar measures with respect to all real numbers on a Cantor dust. We define a local index function on the real numbers for each quasi-self-similar measure at each point in a Cantor dust. The value of the local index function at the real number zero for all the quasi-self-similar measures at each point is the weak local dimension of the point. We also define transformed measures of a quasi-self-similar measure which are closely related to the local index function. We compute the local dimensions of transformed measures of a quasi-self-similar measure to find the multifractal spectrum of the quasi-self-similar measure. Furthermore we give an essential example for the theorem of local dimension of transformed measure. In fact, our result is an ultimate generalization of that of a self-similar measure on a self-similar Cantor set. Furthermore the results also explain the recent results about weak local dimensions on a Cantor dust.