Skip to main content Accessibility help
×
Home

Exact Hausdorff and packing measures of Cantor sets with overlaps

  • HUA QIU (a1)

Abstract

Let $K$ be the attractor of a linear iterated function system (IFS) $S_{j}(x)={\it\rho}_{j}x+b_{j},j=1,\ldots ,m$ , on the real line $\mathbb{R}$ satisfying the generalized finite type condition (whose invariant open set ${\mathcal{O}}$ is an interval) with an irreducible weighted incidence matrix. This condition was recently introduced by Lau and Ngai [A generalized finite type condition for iterated function systems. Adv. Math.208 (2007), 647–671] as a natural generalization of the open set condition, allowing us to include many important overlapping cases. They showed that the Hausdorff and packing dimensions of $K$ coincide and can be calculated in terms of the spectral radius of the weighted incidence matrix. Let ${\it\alpha}$ be the dimension of $K$ . In this paper, we state that

$$\begin{eqnarray}{\mathcal{H}}^{{\it\alpha}}(K\cap J)\leq |J|^{{\it\alpha}}\end{eqnarray}$$
for all intervals $J\subset \overline{{\mathcal{O}}}$ , and
$$\begin{eqnarray}{\mathcal{P}}^{{\it\alpha}}(K\cap J)\geq |J|^{{\it\alpha}}\end{eqnarray}$$
for all intervals $J\subset \overline{{\mathcal{O}}}$ centered in $K$ , where ${\mathcal{H}}^{{\it\alpha}}$ denotes the ${\it\alpha}$ -dimensional Hausdorff measure and ${\mathcal{P}}^{{\it\alpha}}$ denotes the ${\it\alpha}$ -dimensional packing measure. This result extends a recent work of Olsen [Density theorems for Hausdorff and packing measures of self-similar sets. Aequationes Math.75 (2008), 208–225] where the open set condition is required. We use these inequalities to obtain some precise density theorems for the Hausdorff and packing measures of $K$ . Moreover, using these density theorems, we describe a scheme for computing ${\mathcal{H}}^{{\it\alpha}}(K)$ exactly as the minimum of a finite set of elementary functions of the parameters of the IFS. We also obtain an exact algorithm for computing ${\mathcal{P}}^{{\it\alpha}}(K)$ as the maximum of another finite set of elementary functions of the parameters of the IFS. These results extend previous ones by Ayer and Strichartz [Exact Hausdorff measure and intervals of maximum density for Cantor sets. Trans. Amer. Math. Soc.351 (1999), 3725–3741] and by Feng [Exact packing measure of Cantor sets. Math. Natchr.248–249 (2003), 102–109], respectively, and apply to some new classes allowing us to include Cantor sets in $\mathbb{R}$ with overlaps.

Copyright

References

Hide All
[1]Ayer, E. and Strichartz, R.. Exact Hausdorff measure and intervals of maximum density for Cantor sets. Trans. Amer. Math. Soc. 351 (1999), 37253741.
[2]Falconer, K. J.. The Geometry of Fractal Sets (Cambridge Tracts in Mathematics, 85). Cambridge University Press, Cambridge, 1985.
[3]Falconer, K. J.. Dimensions and measures of quasi self-similar sets. Proc. Amer. Math. Soc. 106 (1989), 543554.
[4]Falconer, K. J.. Fractal Geometry (Mathematical Foundations and Applications). John Wiley & Sons, Chichester, 1990.
[5]Feng, D.-J.. Exact packing measure of Cantor sets. Math. Natchr. 248–249 (2003), 102109.
[6]Feng, D.-J., Hua, S. and Wen, Z.-Y.. Some relations between packing premeasure and packing measure. Bull. Lond. Math. Soc. 31 (1999), 665670.
[7]Hutchinson, J. E.. Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), 713747.
[8]Jin, N. and Yau, S. S. T.. General finite type IFS and M-matrix. Comm. Anal. Geom. 13 (2005), 821843.
[9]Lalley, S. P.. 𝛽-expansions with deleted digits for Pisot numbers. Trans. Amer. Math. Soc. 349 (1997), 43554365.
[10]Lau, K.-S. and Ngai, S.-M.. A generalized finite type condition for iterated function systems. Adv. Math. 208 (2007), 647671.
[11]Lau, K.-S. and Wang, X.-Y.. Iterated function systems with a weak separation condition. Stud. Math. 161 (3) (2004), 249268.
[12]Marion, J.. Measure de Hausdorff d’un fractal à similitude interne. Ann. Sci. Math. Québec 10 (1986), 5184.
[13]Marion, J.. Measures de Hausdorff d’ensembles fractals. Ann. Sci. Math. Québec 11 (1987), 111132.
[14]Mattila, P.. Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability. Cambridge University Press, Cambridge, 1995.
[15]Mauldin, R. D. and Williams, S. C.. Hausdorff dimension in graph directed constructions. Trans. Amer. Math. Soc. 309 (1988), 811829.
[16]Mauldin, R. D. and Urbański, M.. Dimensions and measures in infinite iterated function systems. Proc. Lond. Math. Soc. (3) 73 (1996), 105154.
[17]Moran, P. A. P.. Additive functions of intervals and Hausdorff measure. Proc. Camb. Phil. Soc. 42 (1946), 1523.
[18]Morán, M.. Computability of the Hausdorff and pakcing measures on self-similar sets and the self-similar tiling principle. Nonlinearity 18 (2005), 559570.
[19]Ngai, S.-M. and Wang, Y.. Hausdorff dimension of self-similar sets with overlaps. J. Lond. Math. Soc. (2) 63 (2001), 655672.
[20]Olsen, L.. Density theorems for Hausdorff and packing measures of self-similar sets. Aequationes Math. 75 (2008), 208225.
[21]Qiu, H.. Continuity of packing measure function of self-similar iterated function systems. Ergod. Th. & Dynam. Sys. 32(3) (2012), 11011115.
[22]Rao, H. and Wen, Z.-Y.. A class of self-similar fractals with overlap structure. Adv. Appl. Math. 20 (1998), 5072.
[23]Schief, A.. Separation properties for self-similar sets. Proc. Amer. Math. Soc. 122 (1994), 111115.
[24]Zhou, Z.-L. and Feng, L.. Twelve open questions on the exact value of the Hausdorff measure and on the topological entropy: a brief review of recent results. Nonlinearity 17 (2004), 493502.

Exact Hausdorff and packing measures of Cantor sets with overlaps

  • HUA QIU (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed