Hostname: page-component-848d4c4894-mwx4w Total loading time: 0 Render date: 2024-07-01T17:01:44.867Z Has data issue: false hasContentIssue false
  • English
  • Français

Local conical dimensions for measures

Published online by Cambridge University Press:  07 February 2013

DE–JUN FENG ,
ANTTI KÄENMÄKI  and
VILLE SUOMALA
DE–JUN FENG
Affiliation:
The Chinese University of Hong Kong, Shatin, Hong Kong. e-mail: djfeng@math.cuhk.edu.hk
ANTTI KÄENMÄKI
Affiliation:
Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014 University of Jyväskylä, Finland. e-mail: antti.kaenmaki@jyu.fi
VILLE SUOMALA
Affiliation:
Department of Mathematical Sciences, P.O. Box 3000, FI-90014 University of Oulu, Finland. e-mail: ville.suomala@oulu.fi
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the decay of μ(B(x,r)∩C)/μ(B(x,r)) as r ↓ 0 for different kinds of measures μ on ℝn and various cones C around x. As an application, we provide sufficient conditions that imply that the local dimensions can be calculated via cones almost everywhere.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 154 , Issue 3 , May 2013 , pp. 465 - 479
Copyright
Copyright © Cambridge Philosophical Society 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Bandt, C. and Kravchenko, A.Differentiability of fractal curves. Nonlinearity 24 (10) (2011), 2717–2728.CrossRefGoogle Scholar
[2]Besicovitch, A. S.On the fundamental geometrical properties of linearly measurable plane sets of points II. Math. Ann. 115 (1938), 296–329.CrossRefGoogle Scholar
[3]Csörnyei, M., Käenmäki, A., Rajala, T., and Suomala, V.Upper conical density results for general measures on ℝn. Proc. Edinburgh Math. Soc. 53 (2) (2010), 311–331.CrossRefGoogle Scholar
[4]Erdős, P. and Révész, P.On the length of the longest head-run. In Topics in information theory (Second Colloq., Keszthely, 1975) Colloq. Math. Soc. János Bolyai Vol. 16. (North-Holland, Amsterdam, 1977) pages 219–228.Google Scholar
[5]Falconer, K.Fractal Geometry. (John Wiley & Sons Ltd., Chichester, 1990). Mathematical Foundations and Applications.Google Scholar
[6]Falconer, K. J.Geometry of Fractal Sets (Cambridge University Press, 1985).CrossRefGoogle Scholar
[7]Falconer, K. J.One-sided multifractal analysis and points of non-differentiability of devil's staircases. Math. Proc. Camb. Phil. Soc. 136 (1) (2004), 167–174.CrossRefGoogle Scholar
[8]Federer, H.Geometric Measure Theory (Springer-Verlag, Berlin, 1969).Google Scholar
[9]Feng, D.-J. and Hu, H.Dimension theory of iterated function systems. Comm. Pure Appl. Math. 62 (11) (2009), 1435–1500.CrossRefGoogle Scholar
[10]Käenmäki, A.On the geometric structure of the limit set of conformal iterated function systems. Publ. Mat. 47 (1) (2003), 133–141.CrossRefGoogle Scholar
[11]Käenmäki, A.Geometric rigidity of a class of fractal sets. Math. Nachr. 279 (1) (2006), 179–187.CrossRefGoogle Scholar
[12]Käenmäki, A.On upper conical density results. In Benedetto, J. J., Barral, J. and Seuret, S., editors, Recent Developments in Fractals and Related Fields, Applied and Numerical Harmonic Analysis (Birkhäuser Boston, 2010), pages 45–54.CrossRefGoogle Scholar
[13]Käenmäki, A., Rajala, T., and Suomala, V. Local homogeneity and dimensions of measures. preprint, arXiv:1003.2895v2 (2012).Google Scholar
[14]Käenmäki, A. and Suomala, V.Conical upper density theorems and porosity of measures. Adv. Math. 217 (3) (2008), 952–966.CrossRefGoogle Scholar
[15]Käenmäki, A. and Suomala, V.Nonsymmetric conical upper density and k-porosity. Trans. Amer. Math. Soc. 363 (3) (2011), 1183–1195.CrossRefGoogle Scholar
[16]Käenmäki, A. and Vilppolainen, M.Separation conditions on controlled Moran constructions. Fund. Math. 200 (1) (2008), 69–100.CrossRefGoogle Scholar
[17]Hutchinson, J. E.Fractals and self-similarity. Indiana Univ. Math. J. 30 (5) (1981), 713–747.CrossRefGoogle Scholar
[18]Lau, K.-S., Rao, H., and Ye., Y.-L.Corrigendum: Iterated function system and Ruelle operator. J. Math. Anal. Appl. 262 (1) (2001), 446–451.CrossRefGoogle Scholar
[19]Lorent, A.A generalised conical density theorem for unrectifiable sets. Ann. Acad. Sci. Fenn. Math. 28 (2) (2003), 415–431.Google Scholar
[20]Marstrand, J. M.Some fundamental geometrical properties of plane sets of fractional dimensions. Proc. London Math. Soc. (3) 4 (1954), 257–301.CrossRefGoogle Scholar
[21]Martín, M. Á. and Mattila, P.k-dimensional regularity classifications for s-fractals. Trans. Amer. Math. Soc. 305 (1) (1988), 293–315.Google Scholar
[22]Mattila, P.On the structure of self-similar fractals. Ann. Acad. Sci. Fenn. Ser. A I Math. 7 (2) (1982), 189–195.CrossRefGoogle Scholar
[23]Mattila, P.Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability (Cambridge University Press, 1995).CrossRefGoogle Scholar
[24]Mattila, P. and Paramonov, P. V.On geometric properties of harmonic Lip1-capacity. Pacific J. Math. 171 (2) (1995), 469–491.CrossRefGoogle Scholar
[25]Mera, M., Morán, M., Preiss, D., and Zajíček., L.Porosity, σ-porosity and measures. Nonlinearity 16 (1) (2003), 247–255.CrossRefGoogle Scholar
[26]Mera, M. E. and Morán, M.Attainable values for upper porosities of measures. Real Anal. Exchange 26 (1) (2000/01), 101–115.CrossRefGoogle Scholar
[27]Orponen, T. and Sahlsten, T.Tangent measures of non-doubling measures. Math. Proc. Camb. Phil. Soc. 152 (2012), 555–569.CrossRefGoogle Scholar
[28]Peres, Y., Rams, M., Simon, K., and Solomyak, B.Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets. Proc. Amer. Math. Soc. 129 (9) (2001), 2689–2699.CrossRefGoogle Scholar
[29]Preiss, D.Geometry of measures in Rn: distribution, rectifiability, and densities. Ann. of Math. (2) 125 (3) (1987), 537–643.CrossRefGoogle Scholar
[30]Révész, P.Random walk in random and non-random environments (World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, second edition, 2005).CrossRefGoogle Scholar
[31]Sahlsten, T., Shmerkin, P., and Suomala, V.Dimension, entropy and the local distribution of measures. J. London Math. Soc. to appear. (arXiv:1110.6011), doi:10.1112/jlms/jds043.Google Scholar
[32]Schief, A.Separation properties for self-similar sets. Proc. Amer. Math. Soc. 122 (1) (1994), 111–115.CrossRefGoogle Scholar
[33]Suomala, V.On the conical density properties of measures on ℝn. Math. Proc. Camb. Phil. Soc. 138 (2005), 493–512.CrossRefGoogle Scholar
[34]Suomala, V.Upper porous measures on metric spaces. Illinois J. Math. 52 (3) (2008), 967–980.CrossRefGoogle Scholar