Abstract.
We introduce the concept of average homogeneity of a measure by comparing the measure to the uniform distribution in a relatively simple way. This leads to a very general notion which may be regarded as an inverse of porosity. In this paper the emphasis is given to relations between homogeneity and dimensions of measures. First we consider the effect of homogeneity on dimensions by proving an upper bound to the Hausdorff dimension as a function of homogeneity and its order. The opposite question of how dimensions effect homogeneity is solved by giving an upper bound to homogeneity in terms of upper packing dimension. We also illustrate by examples that all our results are the best possible ones.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beliaev, D.B., Smirnov, S.K.: On dimension of porous measures. Math. Ann. 323, 123–141 (2002)
Berlinkov, A., Järvenpää, E.: Porosities in Mandelbrot percolation. preprint 280 http://www.math.jyu.fi/tutkimus/julkaisut.html
Billingsley, P.: Ergodic Theory and Information. John Wiley & Sons, New York, 1965
Cutler, C.D.: Strong and weak duality principles for fractal dimension in Euclidean space. Math. Proc. Cambridge Philos. Soc. 118, 393–410 (1995)
Eckmann, J.-P., Järvenpää, E., Järvenpää, M.: Porosities and dimensions of measures. Nonlinearity 13, 1–18 (2000)
Falconer, K.J.: Techniques in fractal geometry. John Wiley & Sons, Chichester, 1997
Heurteaux, Y.: Estimations de la dimension inférieure et de la dimension supérieure des mesures. Ann. Inst. H. Poincaré Probab. Statist. 34, 309–338 (1998)
Järvenpää, E., Järvenpää, M.: Porous measures on ℝn: local structure and dimensional properties. Proc. Amer. Math. Soc. 130, 419–426 (2001)
Kechris, A.S.: Classical Descriptive Set Theory. Springer Verlag, New York, 1995
Koskela, P., Rohde, S.: Hausdorff dimension and mean porosity. Math. Ann. 309, 593–609 (1997)
Mattila, P.: Distribution of sets and measures along planes. Proc. London Math. Soc. 38(2), 125–132 (1998)
Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces: Fractals and rectifiability. Cambridge University Press, Cambridge, 1995
Mera, M.E., Morán, M.: Attainable values for upper porosities of measures. Real. Anal. Exchange 26, 101–115 (2000/01)
Mera, M.E., Morán, M., Preiss, D., Zajíček, L.: Porosity, σ-porosity and measures. Nonlinearity 16, 247–255 (2003)
Salli, A.: On the Minkowski dimension of strongly porous fractal sets in ℝn. Proc. London Math. Soc. 62(3), 353–372 (1991)
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematics Subject Classification (2000): 28A75, 28A80
MJ acknowledges the support of the Academy of Finland, project #48557.
Rights and permissions
About this article
Cite this article
Järvenpää, E., Järvenpää, M. Average homogeneity and dimensions of measures. Math. Ann. 331, 557–576 (2005). https://doi.org/10.1007/s00208-004-0595-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-004-0595-1