Abstract
Recently Lou and Wu obtained the formulas of pointwise dimensions of some Moran measures on Moran sets in ℝd under the strong separation condition. In this paper, we prove that the result is still true under the open set condition. Due to the lack of the strong separation condition, our approach is essentially different from that used by Lou and Wu. We also obtain the formulas of the Hausdorff and packing dimensions of the Moran measures and discuss some interesting examples.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Billingsley P. Hausdorff dimension in probability theory II. Illinois J Math, 1961, 5: 291–298
Cawley R, Mauldin R D. Multifractal decompositions of Moran fractals. Adv Math, 1992, 92: 196–236
Das M. Billingsley’s packing dimension. Proc Amer Math Soc, 2008, 136: 273–278
Falconer K J. Techniques in Fractal Geometry. Chichester: John Wiley and Sons, 1997
Falconer K J. The Geometry of Fractal Sets. Cambridge: Cambridge University Press, 1985
Geronimo J S, Hardin D P. An exact formula for the measure dimensions associated with a class of piecewise linear maps. Constr Approx, 1989, 5: 89–98
Heurteaux Y. Dimension of measures: the probabilistic approach. Publ Mat, 2007, 51: 243–290
Hutchinson J E. Fractals and self-similarity. Indiana Univ Math J, 1981, 30: 713–747
Li W X. An equivalent definition of packing dimension and its application. Nonlinear Anal, 2009, 10: 1618–1626
Lou M L, Wu M. The pointwise dimensions of Moran measures. Sci China Math, 2010, 53: 1283–1292
Olsen L. A multifractal formalism. Adv Math, 1995, 116: 82–196
Pesin Y B. Dimension Theory in Dynamical Systems: Contemporary Views and Application, Chicago Lectures in Mathematics. Chicago: University of Chicago Press, 1997
Rogers C A. Hausdorff Measures. Cambridge: Cambridge University Press, 1970
Strichartz R S. Self-similar measures and their Fourier transforms. Indiana Univ Math J, 1990, 39: 797–817
Wen Z Y. Mathematical Foundations of Fractal Geometry. Shanghai: Shanghai Scientific and Technological Education Publishing House, 2000
Wen Z Y. Moran sets and Moran classes. Chinese Sci Bull, 2001, 46: 1849–1856
Wen Z Y, Wen Z X. Sequences of substitutions and related topics. Adv Math (China), 1989, 18: 270–293
Wu M. The multifractal spectrum of some Moran measures. Sci China Ser A, 2005, 48: 1097–1112
Wu M. The singularity spectrum f(α) of some Moran fractals. Monatsh Math, 2005, 144: 141–155
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Li, J., Wu, M. Pointwise dimensions of general Moran measures with open set condition. Sci. China Math. 54, 699–710 (2011). https://doi.org/10.1007/s11425-011-4187-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-011-4187-8