Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the boundaries of self-similar tiles in $ \mathbb{R}^1$


Author: Xing-Gang He
Journal: Proc. Amer. Math. Soc. 134 (2006), 3163-3170
MSC (2000): Primary 28A80, 05B45
Published electronically: June 5, 2006
MathSciNet review: 2231899
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The aim of this note is to study the construction of the boundary of a self-similar tile, which is generated by an iterated function system $ \{\phi_i(x)=\frac{1}{N} (x+d_i)\}_{i=1}^N$. We will show that the boundary has complicated structure (no simple points) in general; however, it is a regular fractal set.


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

  • [B] C. BANDT, Self-similar sets 5. Integer matrices and fractal tilings of $ \mathbb{R}^n$, Proc. Amer. Math. Soc. 112(1991), 549-562. MR 1036982 (92d:58093)
  • [F] K. J. FALCONER, Techniques in fractal geometry. John Wiley and Sons, Ltd., Chichester, 1997. MR 1449135 (99f:28013)
  • [GH] K. GR¨OCHENIG AND A. HAAS , Self-similar lattice tilings, J. Fourier Anal. Appl. 1(1994), 131-170. MR 1348740 (96j:52037)
  • [HL] X.G. HE AND K.S. LAU, On a generalized dimension of self-affine fractals, Preprint.
  • [HLR] X.G. HE, K.S. LAU AND H. RAO, Self-affine sets and graph-directed systems, Const. Approx., 19 (2003), no. 3, 373-397. MR 1979057 (2004k:52026)
  • [K] R. KENYON, Self-replicating tilings, Symbolic Dynamics and Its Applications (ed. P. Walters), Contemporary Math., Vol. 135, 1992, 239-264. MR 1185093 (94a:52043)
  • [LR] K.S. LAU AND H. RAO, On one-dimensional self-similar tilings and the $ pq$-tilings. Trans. Amer. Math. Soc., 355(2003), 1401-1414. MR 1946397 (2003k:11033)
  • [LW] J. C. LAGARIAS AND Y. WANG , Tiling the line with the translates of one tile, Invent. Math., 124, fasc. 2 (1996), 341-365. MR 1369421 (96i:05040)
  • [LW1] J. C. LAGARIAS AND Y. WANG , Self-affine tile in $ \mathbb{R}^n$, Adv. Math. 121(1996), 21-49. MR 1399601 (97d:52034)
  • [LW2] J. C. LAGARIAS AND Y. WANG , Integral self-affine tiles in $ \mathbb{R}^n$ I. Standard and non-standard digit sets, J. London Math. Soc., 54(1996), 161-179. MR 1395075 (97f:52031)
  • [LW3] J. C. LAGARIAS AND Y. WANG , Integral self-affine tiles in $ \mathbb{R}^n$ Part II. Lattice tilings, J. Fourier Anal. Appl., 3(1997), 84-102. MR 1428817 (98b:52026)
  • [O] A. M. ODLYZKO, Non-negative digit sets in positional number systems. Proc. London Math. Soc. (3) 37 (1978) 213-229. MR 0507604 (80m:10004)
  • [PSS] Y. PERES, W. SCHLAG AND B. SOLOMYAK, Sixty years of Bernoulli convolutions, Fractals and Stochastics II, (C. Band, S. Graf and M. Zaehle, eds.), Progress in probability 46, 39-65. Birkhäuser, 2000. MR 1785620 (2001m:42020)
  • [S] E. SENETA, Non-negative matrices and Markov chains, Second Edition, Springer-Verlag, 1981. MR 0719544 (85i:60058)
  • [SW] R. S. STRICHARTZ AND Y. WANG, Geometry of self-affine tiles I, Indiana U. Math. J. 48 (1999), no. 1, 1-23. MR 1722192 (2000k:52017)
  • [T] W. THURSTON, Group tilings, and finite state automata, AMS Colloquium Lecture Notes, 1989.
  • [V] A. VINCE, Digit tiling of Euclidean space, Direction in mathematical quasicrystals, CRM Monograph Ser., 13, AMS, Providence RI, 2000, 329-370. MR 1798999 (2002g:52025)
  • [X] Y. XU, Fractals and Tilings, Ph.D. Thesis, University of Pittsburgh 2000.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 28A80, 05B45

Retrieve articles in all journals with MSC (2000): 28A80, 05B45


Additional Information

Xing-Gang He
Affiliation: Department of Mathematics, Central China Normal University, Wuhan, 430079, People’s Republic of China
Email: xingganghe@sina.com

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08643-6
PII: S 0002-9939(06)08643-6
Keywords: Box dimension, Hausdorff dimension, self-similar set, self-similar tile, iterated function system.
Received by editor(s): April 14, 2005
Published electronically: June 5, 2006
Additional Notes: This research was supported in part by SRF for ROCS(SEM)
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.