On the Minkowski measurability of fractals
Author:
K. J. Falconer
Journal:
Proc. Amer. Math. Soc. 123 (1995), 11151124
MSC:
Primary 28A80
MathSciNet review:
1224615
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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 . Second, we use a renewal theory argument to point out that "most" selfsimilar fractals are Minkowski measurable and calculate their Minkowski content.
 [1]
Tim
Bedford, Applications of dynamical systems theory to
fractals—a study of cookiecutter 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
 [2]
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
(81f:58012)
 [3]
K. J. Falconer, Fractal geometryMathematical foundations and applications, Wiley, Chichester, 1990.
 [4]
Herbert
Federer, Geometric measure theory, Die Grundlehren der
mathematischen Wissenschaften, Band 153, SpringerVerlag New York Inc., New
York, 1969. MR
0257325 (41 #1976)
 [5]
Steven
P. Lalley, The packing and covering functions of some selfsimilar
fractals, Indiana Univ. Math. J. 37 (1988),
no. 3, 699–710. MR 962930
(89h:28013), http://dx.doi.org/10.1512/iumj.1988.37.37034
 [6]
Steven
P. Lalley, Renewal theorems in symbolic dynamics, with applications
to geodesic flows, nonEuclidean tessellations and their fractal
limits, Acta Math. 163 (1989), no. 12,
1–55. MR
1007619 (91c:58112), http://dx.doi.org/10.1007/BF02392732
 [7]
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
 [8]
Jun
Kigami and Michel
L. Lapidus, Weyl’s problem for the spectral distribution of
Laplacians on p.c.f.\ selfsimilar fractals, Comm. Math. Phys.
158 (1993), no. 1, 93–125. MR 1243717
(94m:58225)
 [9]
Michel
L. Lapidus, Fractal drum, inverse spectral
problems for elliptic operators and a partial resolution of the WeylBerry
conjecture, Trans. Amer. Math. Soc.
325 (1991), no. 2,
465–529. MR
994168 (91j:58163), http://dx.doi.org/10.1090/S00029947199109941685
 [10]
M.
L. Lapidus, Vibrations of fractal drums, the Riemann hypothesis,
waves in fractal media and the WeylBerry 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
(95g:58247)
 [11]
M. L. Lapidus and H. Maier, The Riemann hypothesis, inverse spectral problem for vibrating fractal strings and the modified WeylBerry conjecture, J. London Math. Soc. (to appear).
 [12]
Michel
L. Lapidus and Carl
Pomerance, Fonction zêta de Riemann et conjecture de
WeylBerry pour les tambours fractals, C. R. Acad. Sci. Paris
Sér. I Math. 310 (1990), no. 6, 343–348
(French, with English summary). MR 1046509
(91d:58248)
 [13]
Michel
L. Lapidus and Carl
Pomerance, The Riemann zetafunction and the onedimensional
WeylBerry conjecture for fractal drums, Proc. London Math. Soc. (3)
66 (1993), no. 1, 41–69. MR 1189091
(93k:58217), http://dx.doi.org/10.1112/plms/s366.1.41
 [14]
David
Ruelle, Repellers for real analytic maps, Ergodic Theory
Dynamical Systems 2 (1982), no. 1, 99–107. MR 684247
(84f:58095)
 [1]
 T. Bedford, Applications of dynamical systems theory to fractalsa study of cookie cutter Cantor sets, Fractal Analysis and Geometry, Kluwer Academic, Dordrecht, 1991. MR 1140719
 [2]
 M. V. Berry, Some geometric aspects of wave motion: wavefront dislocations, diffraction catastrophes, diffractals, Geometry of the Laplace Operator, Proc. Sympos. Pure Math., vol. 36, Amer. Math. Soc., Providence, RI, 1980, pp. 1338. MR 573427 (81f:58012)
 [3]
 K. J. Falconer, Fractal geometryMathematical foundations and applications, Wiley, Chichester, 1990.
 [4]
 H. Federer, Geometric measure theory, SpringerVerlag, Berlin, 1969. MR 0257325 (41:1976)
 [5]
 S. P. Lalley, The packing and covering functions of some selfsimilar fractals, Indiana Math. J. 37 (1988), 699710. MR 962930 (89h:28013)
 [6]
 , Renewal theorems in symbolic dynamics, with applications to geodesic flow, noneuclidean tessellations and their fractal limits, Acta Math. 163 (1989), 155. MR 1007619 (91c:58112)
 [7]
 , Probabilistic methods in certain counting problems of ergodic theory, Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, Oxford Univ. Press, Oxford, 1991, pp. 223258. MR 1130178
 [8]
 J. Kigami and M. L. Lapidus, Weyl's problem for the spectral distribution of Laplacians on p.c.f. selfsimilar fractals, Comm. Math. Phys. 158 (1993), 93125. MR 1243717 (94m:58225)
 [9]
 M. L. Lapidus, Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the WeylBerry conjecture, Trans. Amer. Math. Soc. 325 (1991), 465529. MR 994168 (91j:58163)
 [10]
 , Vibrations of fractal drums, the Riemann hypothesis, waves in fractal media, and the WeylBerry conjecture, Ordinary and Partial Differential Equations IV, Longman Sci. Tech., Essex, 1993, pp. 126209. MR 1234502 (95g:58247)
 [11]
 M. L. Lapidus and H. Maier, The Riemann hypothesis, inverse spectral problem for vibrating fractal strings and the modified WeylBerry conjecture, J. London Math. Soc. (to appear).
 [12]
 M. L. Lapidus and C. Pomerance, Fonction zêta de Riemann et conjecture de WeylBerry par les tambours fractals, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), 343348. MR 1046509 (91d:58248)
 [13]
 , The Riemann zetafunction and the onedimensional WeylBerry conjecture for fractal drums, Proc. London Math. Soc. (3) 66 (1993), 4169. MR 1189091 (93k:58217)
 [14]
 D. Ruelle, Repellers for real analytic maps, Ergodic Theory Dynamical Systems 2 (1982), 99109. MR 684247 (84f:58095)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199512246154
PII:
S 00029939(1995)12246154
Article copyright:
© Copyright 1995
American Mathematical Society
