On the boundaries of selfsimilar tiles in
Author:
XingGang He
Journal:
Proc. Amer. Math. Soc. 134 (2006), 31633170
MSC (2000):
Primary 28A80, 05B45
Published electronically:
June 5, 2006
MathSciNet review:
2231899
Fulltext 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 selfsimilar tile, which is generated by an iterated function system . We will show that the boundary has complicated structure (no simple points) in general; however, it is a regular fractal set.
 [B]
C. BANDT, Selfsimilar sets 5. Integer matrices and fractal tilings of , Proc. Amer. Math. Soc. 112(1991), 549562. MR 1036982 (92d:58093)
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K. J. FALCONER, Techniques in fractal geometry. John Wiley and Sons, Ltd., Chichester, 1997. MR 1449135 (99f:28013)
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K. GR¨OCHENIG AND A. HAAS , Selfsimilar lattice tilings, J. Fourier Anal. Appl. 1(1994), 131170. MR 1348740 (96j:52037)
 [HL]
X.G. HE AND K.S. LAU, On a generalized dimension of selfaffine fractals, Preprint.
 [HLR]
X.G. HE, K.S. LAU AND H. RAO, Selfaffine sets and graphdirected systems, Const. Approx., 19 (2003), no. 3, 373397. MR 1979057 (2004k:52026)
 [K]
R. KENYON, Selfreplicating tilings, Symbolic Dynamics and Its Applications (ed. P. Walters), Contemporary Math., Vol. 135, 1992, 239264. MR 1185093 (94a:52043)
 [LR]
K.S. LAU AND H. RAO, On onedimensional selfsimilar tilings and the tilings. Trans. Amer. Math. Soc., 355(2003), 14011414. 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), 341365. MR 1369421 (96i:05040)
 [LW1]
J. C. LAGARIAS AND Y. WANG , Selfaffine tile in , Adv. Math. 121(1996), 2149. MR 1399601 (97d:52034)
 [LW2]
J. C. LAGARIAS AND Y. WANG , Integral selfaffine tiles in I. Standard and nonstandard digit sets, J. London Math. Soc., 54(1996), 161179. MR 1395075 (97f:52031)
 [LW3]
J. C. LAGARIAS AND Y. WANG , Integral selfaffine tiles in Part II. Lattice tilings, J. Fourier Anal. Appl., 3(1997), 84102. MR 1428817 (98b:52026)
 [O]
A. M. ODLYZKO, Nonnegative digit sets in positional number systems. Proc. London Math. Soc. (3) 37 (1978) 213229. 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, 3965. Birkhäuser, 2000. MR 1785620 (2001m:42020)
 [S]
E. SENETA, Nonnegative matrices and Markov chains, Second Edition, SpringerVerlag, 1981. MR 0719544 (85i:60058)
 [SW]
R. S. STRICHARTZ AND Y. WANG, Geometry of selfaffine tiles I, Indiana U. Math. J. 48 (1999), no. 1, 123. 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, 329370. MR 1798999 (2002g:52025)
 [X]
Y. XU, Fractals and Tilings, Ph.D. Thesis, University of Pittsburgh 2000.
 [B]
 C. BANDT, Selfsimilar sets 5. Integer matrices and fractal tilings of , Proc. Amer. Math. Soc. 112(1991), 549562. 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 , Selfsimilar lattice tilings, J. Fourier Anal. Appl. 1(1994), 131170. MR 1348740 (96j:52037)
 [HL]
 X.G. HE AND K.S. LAU, On a generalized dimension of selfaffine fractals, Preprint.
 [HLR]
 X.G. HE, K.S. LAU AND H. RAO, Selfaffine sets and graphdirected systems, Const. Approx., 19 (2003), no. 3, 373397. MR 1979057 (2004k:52026)
 [K]
 R. KENYON, Selfreplicating tilings, Symbolic Dynamics and Its Applications (ed. P. Walters), Contemporary Math., Vol. 135, 1992, 239264. MR 1185093 (94a:52043)
 [LR]
 K.S. LAU AND H. RAO, On onedimensional selfsimilar tilings and the tilings. Trans. Amer. Math. Soc., 355(2003), 14011414. 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), 341365. MR 1369421 (96i:05040)
 [LW1]
 J. C. LAGARIAS AND Y. WANG , Selfaffine tile in , Adv. Math. 121(1996), 2149. MR 1399601 (97d:52034)
 [LW2]
 J. C. LAGARIAS AND Y. WANG , Integral selfaffine tiles in I. Standard and nonstandard digit sets, J. London Math. Soc., 54(1996), 161179. MR 1395075 (97f:52031)
 [LW3]
 J. C. LAGARIAS AND Y. WANG , Integral selfaffine tiles in Part II. Lattice tilings, J. Fourier Anal. Appl., 3(1997), 84102. MR 1428817 (98b:52026)
 [O]
 A. M. ODLYZKO, Nonnegative digit sets in positional number systems. Proc. London Math. Soc. (3) 37 (1978) 213229. 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, 3965. Birkhäuser, 2000. MR 1785620 (2001m:42020)
 [S]
 E. SENETA, Nonnegative matrices and Markov chains, Second Edition, SpringerVerlag, 1981. MR 0719544 (85i:60058)
 [SW]
 R. S. STRICHARTZ AND Y. WANG, Geometry of selfaffine tiles I, Indiana U. Math. J. 48 (1999), no. 1, 123. 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, 329370. MR 1798999 (2002g:52025)
 [X]
 Y. XU, Fractals and Tilings, Ph.D. Thesis, University of Pittsburgh 2000.
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Additional Information
XingGang 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/S0002993906086436
PII:
S 00029939(06)086436
Keywords:
Box dimension,
Hausdorff dimension,
selfsimilar set,
selfsimilar 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.
