On the Minkowski measurability of fractals
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- by K. J. Falconer PDF
- Proc. Amer. Math. Soc. 123 (1995), 1115-1124 Request permission
Abstract:
This note addresses two aspects of Minkowski measurability. First we present a short "dynamical systems" proof of the characterization of Minkowski measurable compact subsets of $\mathbb {R}$. Second, we use a renewal theory argument to point out that "most" self-similar fractals are Minkowski measurable and calculate their Minkowski content.References
- Tim Bedford, Applications of dynamical systems theory to fractals—a study of cookie-cutter Cantor sets, Fractal geometry and analysis (Montreal, PQ, 1989) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 346, Kluwer Acad. Publ., Dordrecht, 1991, pp. 1–44. MR 1140719, DOI 10.1098/rspa.1975.0163
- M. V. Berry, Some geometric aspects of wave motion: wavefront dislocations, diffraction catastrophes, diffractals, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 13–28. MR 573427 K. J. Falconer, Fractal geometry—Mathematical foundations and applications, Wiley, Chichester, 1990.
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
- Steven P. Lalley, The packing and covering functions of some self-similar fractals, Indiana Univ. Math. J. 37 (1988), no. 3, 699–710. MR 962930, DOI 10.1512/iumj.1988.37.37034
- Steven P. Lalley, Renewal theorems in symbolic dynamics, with applications to geodesic flows, non-Euclidean tessellations and their fractal limits, Acta Math. 163 (1989), no. 1-2, 1–55. MR 1007619, DOI 10.1007/BF02392732
- Steven P. Lalley, Probabilistic methods in certain counting problems of ergodic theory, Ergodic theory, symbolic dynamics, and hyperbolic spaces (Trieste, 1989) Oxford Sci. Publ., Oxford Univ. Press, New York, 1991, pp. 223–258. MR 1130178
- Jun Kigami and Michel L. Lapidus, Weyl’s problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals, Comm. Math. Phys. 158 (1993), no. 1, 93–125. MR 1243717
- Michel L. Lapidus, Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture, Trans. Amer. Math. Soc. 325 (1991), no. 2, 465–529. MR 994168, DOI 10.1090/S0002-9947-1991-0994168-5
- M. L. Lapidus, Vibrations of fractal drums, the Riemann hypothesis, waves in fractal media and the Weyl-Berry conjecture, Ordinary and partial differential equations, Vol. IV (Dundee, 1992) Pitman Res. Notes Math. Ser., vol. 289, Longman Sci. Tech., Harlow, 1993, pp. 126–209. MR 1234502 M. L. Lapidus and H. Maier, The Riemann hypothesis, inverse spectral problem for vibrating fractal strings and the modified Weyl-Berry conjecture, J. London Math. Soc. (to appear).
- Michel L. Lapidus and Carl Pomerance, Fonction zêta de Riemann et conjecture de Weyl-Berry pour les tambours fractals, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), no. 6, 343–348 (French, with English summary). MR 1046509
- Michel L. Lapidus and Carl Pomerance, The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums, Proc. London Math. Soc. (3) 66 (1993), no. 1, 41–69. MR 1189091, DOI 10.1112/plms/s3-66.1.41
- David Ruelle, Repellers for real analytic maps, Ergodic Theory Dynam. Systems 2 (1982), no. 1, 99–107. MR 684247, DOI 10.1017/s0143385700009603
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 123 (1995), 1115-1124
- MSC: Primary 28A80
- DOI: https://doi.org/10.1090/S0002-9939-1995-1224615-4
- MathSciNet review: 1224615