Minkowski content and local Minkowski content for a class of self-conformal sets

被引：0
作者：
Uta Freiberg
Sabrina Kombrink
机构：
[1] Universität Siegen,
[2] FB 6—Mathematik,undefined
[3] Universität Bremen,undefined
[4] FB 3—Mathematik,undefined
来源：
Geometriae Dedicata | 2012年 / 159卷
关键词：
Minkowski content; Conformal iterated function system; Self-conformal set; Fractal curvature measures; 28A80; 28A75;
D O I：
暂无
中图分类号：
学科分类号：
摘要：
We investigate (local) Minkowski measurability of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {C}^{1+\alpha}}$$\end{document} images of self-similar sets. We show that (local) Minkowski measurability of a self-similar set K implies (local) Minkowski measurability of its image F and provide an explicit formula for the (local) Minkowski content of F in this case. A counterexample is presented which shows that the converse is not necessarily true. That is, F can be Minkowski measurable although K is not. However, we obtain that an average version of the (local) Minkowski content of both K and F always exists and also provide an explicit formula for the relation between the (local) average Minkowski contents of K and F.
引用
收藏
页码：307 / 325
页数：18
相关论文
共 15 条
  • [1] Falconer K.(1995)On the Minkowski measurability of fractals Proc. Am. Math. Soc. 123 1115-1124
  • [2] Falconer K.(2011)Dixmier traces and coarse multifractal analysis Ergodic Theory Dyn. Syst. 31 369-381
  • [3] Samuel T.(2000)Lacunarity of self-similar and stochastically self-similar sets Trans. Am. Math. Soc. 352 1953-1983
  • [4] Gatzouras D.(2003)Dimensions and singular traces for spectral triples, with applications to fractals J. Funct. Anal. 203 362-400
  • [5] Guido D.(1991)Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture Trans. Am. Math. Soc. 325 465-529
  • [6] Isola T.(1993)The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums Proc. Lond. Math. Soc. (3) 66 41-69
  • [7] Lapidus M.(1996)Spectral asymptotics, renewal theorem, and the Berry conjecture for a class of fractals Proc. Lond. Math. Soc. III. Ser. 72 188-214
  • [8] Lapidus M.(1996)Dimensions and measures in infinite iterated function systems Proc. Lond. Math. Soc. III. Ser. 73 105-154
  • [9] Pomerance C.(1994)Separation properties for self-similar sets Proc. Am. Math. Soc. 122 111-115
  • [10] Levitin M.(2011)Lipschitz-Killing curvatures of self-similar random fractals Trans. Am. Math. Soc. 363 2663-2684
12