The upper densities of symmetric perfect sets

Abstract

In this paper, we give the exact upper densities of Hausdorff measures of a class of symmetric perfect sets.

A mixed multifractal analysis for quasi Ahlfors vector-valued measures

“…Such a mixed analysis has been generating a great attention recently and thus proved to be powerful in describing the local behavior of measures especially fractal ones. (See [9], [10], [11], [49], [50], [52], [53], [54], [60], [69], [70], [71], [73], [74], [75], [76]).…”

“…With these assumptions, it becomes natural to extend the multifractal analysis to the cases introduced in [2] and [20]. The readers may refer to [13], [21], [48], [49], [50], [60], [69], [70], [71], [72], [72], [73], [74], [75], [76] for more understanding.…”

Packing dimension and measure of homogeneous Cantor sets

“…One of the disadvantages of studying fractal measures and dimensions ( Hausdorff and packing measures ) is that in many cases they are hard to calculate or estimate computationally. However, many researchers [1,3,4,5,6,9,10] have found exact values or estimated the lower and upper bounds of measures and dimensions for some fractal sets. In these papers and the references therein the interested reader can find an analysis of the exact dimension and measure of some particular fractal sets.…”

“…In these papers and the references therein the interested reader can find an analysis of the exact dimension and measure of some particular fractal sets. In particular, explicit formulae for Hausdorff dimensions of specific Cantor sets were found in [4,5,6], while in [1,3,9,10] exact values were calculated for Hausdorff measures for some Cantor sets, including homogeneous Cantor sets.…”

“…(iii) For any k ^ 1 and a G D k , 1 ^ j ^ n k , we have where \A\ denote the diameter of A. available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S000497270004048XLet Ek = U I a , E = f) Ek-We call E the homogeneous Cantor set determined by {nk}k^i{Tk}k^i and call J> = {/, : a e £>*} the kth -order basic intervals of E. The middle-third Cantor set and the symmetric perfect set ([5,10]) are well-known examples of homogeneous Cantor sets.2. PACKING DIMENSION OF HOMOGENEOUS C A N T O R SETSIn this section, we express the packing dimension of a homogeneous Cantor set as the explicit form with n* and r^.…”