# Article

MSC: 28A80
Full entry | Fulltext not available (moving wall 24 months)
Keywords:
homogeneous Moran set; $\{m_{k}\}$-Moran set; $\{m_{k}\}$-quasi homogeneous Cantor set; Hausdorff dimension
Summary:
We construct a class of special homogeneous Moran sets, called $\{m_{k}\}$-quasi homogeneous Cantor sets, and discuss their Hausdorff dimensions. By adjusting the value of $\{m_{k}\}_{k\ge 1}$, we constructively prove the intermediate value theorem for the homogeneous Moran set. Moreover, we obtain a sufficient condition for the Hausdorff dimension of homogeneous Moran sets to assume the minimum value, which expands earlier works.
References:
[1] Falconer, K.: Fractal Geometry. Mathematical Foundations and Applications. John Wiley & Sons, Chichester (1990). MR 1102677
[2] Feng, D.-J.: The limited Rademacher functions and Bernoulli convolutions associated with Pisot numbers. Adv. Math. 195 (2005), 24-101. DOI 10.1016/j.aim.2004.06.011 | MR 2145793
[3] Feng, D., Wen, Z., Wu, J.: Some dimensional results for homogeneous Moran sets. Sci. China, Ser. A 40 (1997), 475-482. DOI 10.1007/BF02896955 | MR 1461002
[4] Li, J., Wu, M.: Pointwise dimensions of general Moran measures with open set condition. Sci. China, Math. 54 (2011), 699-710. DOI 10.1007/s11425-011-4187-8 | MR 2786709
[5] Peng, F., Wen, S.: Fatness and thinness of uniform Cantor sets for doubling measures. Sci. China, Math. 54 (2011), 75-81. DOI 10.1007/s11425-010-4148-7 | MR 2764786
[6] Rao, H., Ruan, H.-J., Wang, Y.: Lipschitz equivalence of Cantor sets and algebraic properties of contraction ratios. Trans. Am. Math. Soc. 364 (2012), 1109-1126. DOI 10.1090/S0002-9947-2011-05327-4 | MR 2869169
[7] Wang, B.-W., Wu, J.: Hausdorff dimension of certain sets arising in continued fraction expansions. Adv. Math. 218 (2008), 1319-1339. DOI 10.1016/j.aim.2008.03.006 | MR 2419924
[8] Wang, X. H., Wen, S. Y.: Doubling measures on Cantor sets and their extensions. Acta Math. Hung. 134 (2012), 431-438. DOI 10.1007/s10474-011-0186-z | MR 2886217
[9] Wu, J.: On the sum of degrees of digits occurring in continued fraction expansions of Laurent series. Math. Proc. Camb. Philos. Soc. 138 (2005), 9-20. DOI 10.1017/S0305004104008163 | MR 2127223
[10] Wu, M.: The singularity spectrum {$f(\alpha)$} of some Moran fractals. Monatsh. Math. 144 (2005), 141-155. DOI 10.1007/s00605-004-0254-3 | MR 2123961

Partner of