The L~q Dimension And Entropy Dimension Of Some Moran Measure

The characteristics of fractal sets are frequently demonstrated by the measure but not merely by the set. The Lq dimension of Fractal measures are closely related to the multi-fractal spectrum of measures, and multifractal analysis has been proved to be a very useful technique in the analysis of singular measures, both in theory and applications, but strict proof of a measure to meet the multifractal mechanism or computing the multifractal spectrum are very difficult, so it is important to compute the Lq dimension of fractal measures in the study of multifractal spectrum.Although the Lq dimension of fractal measures frequently appear in physics literature, the strict general results are rarely. We already know, for the self-conformal measures, the literature [19] proved that the Lq dimension and entropy dimension exist; for the self-similar measure and Moran measure, the literatures [15,23,27,31,24] studied their multifractal spectrum, the Legendre spectrum and q-Renyi dimension, but no precision result about the Lq dimension and entropy dimension.For q>0, the Renyi dimension and Hentschel-Procaccia dimension of fractal measures are equivalent, in this paper, with the equivalent of two dimensions of measures, using the skills of reference [25], we prove that the existence of Lq dimension of a kind of Moran mea-sure defined in reference [3] under certain discrete conditions, and give exact expression, for the entropy dimension of the measure, give an estimated range. Further, we compute the Lq dimension of a kind of general Moran measure.