Abstract
An explicit formula of the packing dimension of probability measures associated with \({\widetilde{Q}}\)-representation is given and the packing dimension of the supports of the corresponding measures is determined.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Albeverio, V. Koshmanenko, M. Pratsiovytyi and G. Torbin, \({\widetilde{Q}}\)-representation of real numbers and fractal probability distributions, http://arxiv.org/abs/math/0308007.
Albeverio S., Torbin G.: Fractal properties of singular probability distributions with independent Q *-digits. Bull. Sci. Math. 129, 357–367 (2005)
Das M.: Billingsley’s packing dimension. Proc. Amer. Math. Soc. 136, 273–278 (2008)
Edgar G.A.: Integral, Probability and Fractal Measures. Springer, New York (1997)
Falconer K.J.: Techniques in Fractal Geometry. John Wiley and Sons, Chichester (1997)
Falconer K.J.: The Geometry of Fractal Sets. Cambridge University Press, Cambridge (1985)
Falconer K.J.: Fractal Geometry-Mathematical Foundations and Applications. John Wiley and Sons Ltd, Chichester (1990)
Geronimo J.S., Hardin D.P.: An exact formula for the measure dimensions associated with a class of piecewise linear maps. Constr. Approx. 5, 89–98 (1989)
Heurteaux Y.: Dimension of measures: the probabilistic approach. Publ. Mat. 51, 243–290 (2007)
J. J. Li, Hausdorff dimension of measures associated with \({\widetilde{Q}}\)-representation, Bull. Math. Soc. Sci. Math. Roumanie. submitted.
Li W.X.: An equivalent definition of packing dimension and its application. Nonlinear Anal. 10, 1618–1626 (2009)
Jaroszewska J., Rams M.: On the Hausdorff dimension of invariant measures of weakly contracting on average measurable IFS. J. Stat. Phys. 132, 907–919 (2008)
Jordan T., Pollicott M.: The Hausdorff dimension of measures for iterated function systems which contract on average. Discrete Contin. Dynam. Systems. 22, 235–246 (2008)
Mihail A., Miculescu R.: A Generalization of the Hutchinson Measure. Mediterr. J. Math. 6, 203–213 (2009)
Myjak J., Szarek T.: On Hausdorff dimension of invariant measures arising from non-contractive iterated function systems. Ann. Mat. Pura Appl. 181, 223–237 (2002)
C. A. Rogers, Hausdorff measures. Cambridge University Press, 1970.
Torbin G.: Probability distributions with independent Q-symbols and transformations preserving the Hausdorff dimension. Theory of Stochastic Processes 29, 291–293 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Li, J. Packing Dimension of Measures Associated with \({\widetilde{Q}}\)-Representation. Mediterr. J. Math. 9, 655–668 (2012). https://doi.org/10.1007/s00009-011-0138-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00009-011-0138-4