Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

   
 
 

 

Self-similar sets with an open set condition and great variety of overlaps


Authors: Christoph Bandt and Nguyen Viet Hung
Journal: Proc. Amer. Math. Soc. 136 (2008), 3895-3903
MSC (2000): Primary 28A80; Secondary 37B10, 37F20
DOI: https://doi.org/10.1090/S0002-9939-08-09349-0
Published electronically: May 22, 2008
MathSciNet review: 2425729
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For a very simple family of self-similar sets with two pieces, we prove, using a technique of Solomyak, that the intersection of the pieces can be a Cantor set with any dimension in $ [0,0.2]$ as well as a finite set of any cardinality $ 2^m$. The main point is that the open set condition is fulfilled for all these examples.


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

  • 1. C. Bandt, Self-similar measures, Ergodic Theory, Analysis and Efficient Simulation of Dynamical Systems (ed. B. Fiedler), Springer, 2001, 31-46. MR 1850300 (2002j:28011)
  • 2. C. Bandt, On the Mandelbrot set for pairs of linear maps, Nonlinearity 15 (2002), 1127-1147. MR 1912290 (2004f:28010)
  • 3. C. Bandt and S. Graf, Self-similar sets VII. A characterization of self-similar fractals with positive Hausdorff measure. Proc. Amer. Math. Soc. 114 (1992), 995-1001. MR 1100644 (93d:28014)
  • 4. C. Bandt, N.V. Hung and H. Rao, On the open set condition for self-similar fractals, Proc. Amer. Math. Soc. 134 (2005), 1369-1374. MR 2199182 (2006m:28007)
  • 5. C. Bandt and H. Rao, Topology and separation of self-similar fractals in the plane, Nonlinearity 20 (2007), 1463-1474. MR 2327133
  • 6. M.F. Barnsley, Fractals Everywhere, 2nd ed., Academic Press, 1993. MR 1231795 (94h:58101)
  • 7. K.J. Falconer, Fractal Geometry, Wiley, 1990. MR 1102677 (92j:28008)
  • 8. M. Hata, On the structure of self-similar sets, Japan J. Appl. Math. 2 (1985), 281-414. MR 839336 (87g:58080)
  • 9. J.E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713-747. MR 625600 (82h:49026)
  • 10. S.M. Ngai and Y. Wang, Hausdorff dimension of overlapping self-similar sets. J. London Math. Soc. 63 (2001), 655-672. MR 1825981 (2002c:28010)
  • 11. A. Schief, Separation properties for self-similar sets. Proc. Amer. Math. Soc. 122 (1994), 111-115. MR 1191872 (94k:28012)
  • 12. N. Sidorov, Combinatorics of linear iterated function systems with overlaps, Nonlinearity 20 (2007), 1290-1312. MR 2312394
  • 13. B. Solomyak, `Mandelbrot set' for pairs of linear maps: the local geometry, Analysis in Theory and Applications 20:2 (2004), 149-157. MR 2095457 (2005i:28022)
  • 14. B. Solomyak, On the `Mandelbrot set' for pairs of linear maps: asymptotic self-similarity, Nonlinearity 18 (2005), 1927-1943. MR 2164725 (2006d:37086)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 28A80, 37B10, 37F20

Retrieve articles in all journals with MSC (2000): 28A80, 37B10, 37F20


Additional Information

Christoph Bandt
Affiliation: Institute for Mathematics and Informatics, Arndt University, 17487 Greifswald, Germany
Email: bandt@uni-greifswald.de

Nguyen Viet Hung
Affiliation: Department of Mathematics, Hue University, Hue, Vietnam
Email: nvh0@yahoo.com

DOI: https://doi.org/10.1090/S0002-9939-08-09349-0
Received by editor(s): March 16, 2007
Received by editor(s) in revised form: September 19, 2007
Published electronically: May 22, 2008
Communicated by: Jane M. Hawkins
Article copyright: © Copyright 2008 By the authors

American Mathematical Society