## Arkiv för Matematik

• Ark. Mat.
• Volume 48, Number 1 (2010), 149-176.

### General Hausdorff functions, and the notion of one-sided measure and dimension

Claude Tricot

#### Abstract

The main facts about Hausdorff and packing measures and dimensions of a Borel set E are revisited, using determining set functions $\phi_\alpha\colon\mathcal{B}_E\to(0,\infty)$, where $\mathcal{B}_E$ is the family of all balls centred on E and α is a real parameter. With mild assumptions on φα, we verify that the main density results hold, as well as the basic properties of the corresponding box dimension. Given a bounded open set V in ℝD, these notions are used to introduce the interior and exterior measures and dimensions of any Borel subset of ∂V. We stress that these dimensions depend on the choice of φα. Two determining functions are considered, φα(B)=VolD(BV)diam(B)α-D and φα(B)=VolD(BV)α/D, where VolD denotes the D-dimensional volume.

#### Article information

Source
Ark. Mat., Volume 48, Number 1 (2010), 149-176.

Dates
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.afm/1485907100

Digital Object Identifier
doi:10.1007/s11512-008-0087-8

Mathematical Reviews number (MathSciNet)
MR2594591

Zentralblatt MATH identifier
1196.28010

Rights

#### Citation

Tricot, Claude. General Hausdorff functions, and the notion of one-sided measure and dimension. Ark. Mat. 48 (2010), no. 1, 149--176. doi:10.1007/s11512-008-0087-8. https://projecteuclid.org/euclid.afm/1485907100

#### References

• Billingsley, P., Ergodic Theory and Information, Wiley, New York, 1965.
• Falconer, K. J., The Geometry of Fractal Sets, Cambridge Tracts in Mathematics 85, Cambridge University Press, Cambridge, 1986.
• Grebogi, C., McDonald, S. W., Ott, E. and Yorke, J. A., Exterior dimension of fat fractals, Phys. Lett. A 110 (1985), 1–4; correction in Phys. Lett. A 113 (1986), 495.
• Hausdorff, F., Dimension und äußeres Maß, Math. Ann. 79 (1918), 157–179.
• Heurteaux, Y. and Jaffard, S., Multifractal Analysis of Images: New connexions between Analysis and Geometry, in Proc. of the NATO-ASI Conference on Imaging for Detection and Identification, pp. 169–194, Springer, Berlin–Heidelberg, 2008.
• Morse, A. P. and Randolph, J. F., The φ rectifiable subsets of the plane, Trans. Amer. Math. Soc. 55 (1944), 236–305.
• Olsen, L., A multifractal formalism, Adv. Math. 116 (1995), 82–196.
• Peyrière, J., A vectorial multifractal formalism, in Fractal Geometry and Applications: a Jubilee of Benoît Mandelbrot, Proc. Sympos. Pure Math. 72, pp. 217–230, Amer. Math. Soc., Providence, RI, 2004.
• Saint Raymond, X. and Tricot, C., Packing regularity of sets in n-space, Math. Proc. Cambridge Philos. Soc. 103 (1988), 133–145.
• Taylor, S. J. and Tricot, C., Packing measure, and its evaluation for a Brownian path, Trans. Amer. Math. Soc. 288 (1985), 679–699.
• Taylor, S. J. and Tricot, C., The packing measure of rectifiable subsets of the plane, Math. Proc. Cambridge Philos. Soc. 99 (1986), 285–296.
• Tricot, C., Two definitions of fractional dimension, Math. Proc. Cambridge Philos. Soc. 91 (1982), 57–74.
• Tricot, C., The geometry of the complement of a fractal set, Phys. Lett. A 114 (1986), 430–434.
• Tricot, C., Dimensions aux bords d’un ouvert, Ann. Sci. Math. Québec 11 (1987), 205–235.
• Tricot, C., Curves and Fractal Dimension, Springer, New York, 1995.