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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The Hausdorff dimension of visible sets of planar continua

Author: Toby C. O'Neil
Journal: Trans. Amer. Math. Soc. 359 (2007), 5141-5170
MSC (2000): Primary 28A80; Secondary 28A78, 31A15.
Published electronically: June 4, 2007
MathSciNet review: 2327025
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Abstract | References | Similar Articles | Additional Information

Abstract: For a compact set $ \Gamma\subset\mathbb{R}^2$ and a point $ x$, we define the visible part of $ \Gamma$ from $ x$ to be the set

$\displaystyle \Gamma_x=\{u\in \Gamma: [x,u]\cap \Gamma=\{u\}\}.$

(Here $ [x,u]$ denotes the closed line segment joining $ x$ to $ u$.)

In this paper, we use energies to show that if $ \Gamma$ is a compact connected set of Hausdorff dimension greater than one, then for (Lebesgue) almost every point $ x\in\mathbb{R}^2$, the Hausdorff dimension of $ \Gamma_x$ is strictly less than the Hausdorff dimension of $ \Gamma$. In fact, for almost every $ x$,

$\displaystyle \dim_H (\Gamma_x)\leq \frac{1}{2}+\sqrt{\dim_H(\Gamma)-\frac{3}{4}}.$

We also give an estimate of the Hausdorff dimension of those points where the visible set has dimension greater than $ \sigma+\frac{1}{2}+\sqrt{\dim_H (\Gamma)-\frac{3}{4}}$ for some $ \sigma>0$.

References [Enhancements On Off] (What's this?)

  • 1. K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. MR 867284
  • 2. H. Federer,
    Geometric Measure Theory, Springer-Verlag (1996).
  • 3. Esa Järvenpää, Maarit Järvenpää, Paul MacManus, and Toby C. O’Neil, Visible parts and dimensions, Nonlinearity 16 (2003), no. 3, 803–818. MR 1975783,
  • 4. Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597
  • 5. K. Kuratowski, Topology. Vol. I, New edition, revised and augmented. Translated from the French by J. Jaworowski, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966. MR 0217751
  • 6. Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890
  • 7. O. Nikodym,
    Sur les points linéairement accessibles des ensembles plans, Fundamenta Mathematicae 7 (1925), 250-258.
  • 8. O. Nikodym,
    Sur la mesure des ensembles plans dont tous les points sont rectilinéairement accessibles, Fundamenta Mathematicae 10 (1927), 116-168.
  • 9. O. Nikodym,
    Sur un ensemble plan et fermé dont les points qui sont rectilinéairement accessibles forment un ensemble non mesurable ($ B$), Fundamenta Mathematicae 11 (1928), 239-263.
  • 10. C. A. Rogers, Hausdorff measures, Cambridge University Press, London-New York, 1970. MR 0281862
  • 11. P. Urysohn,
    Problème 29, Fundamenta Mathematicae 5 (1923), 337.
  • 12. P. Urysohn,
    Sur les points accessibles des ensembles fermés, Proceedings Amsterdam 28 (1925), 984-993.

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Additional Information

Toby C. O'Neil
Affiliation: Faculty of Mathematics and Computing, The Open University, Walton Hall, Milton Keynes, MK7 6AA, United Kingdom

Keywords: Visible sets, Hausdorff dimension
Received by editor(s): November 6, 2003
Published electronically: June 4, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.