Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Home
  • Home
Hostname: page-component-559fc8cf4f-s5ss2 Total loading time: 0.443 Render date: 2021-02-26T21:23:45.336Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }
EnglishFrançais
Nagoya Mathematical Journal Nagoya Mathematical Journal

Article contents

BOUNDARY PARAMETRIZATION AND THE TOPOLOGY OF TILES

Part of: Classical measure theory Discrete geometry

Published online by Cambridge University Press:  19 October 2016

SHIGEKI AKIYAMA  and
BENOÎT LORIDANT
SHIGEKI AKIYAMA
Affiliation:
Institute of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki, Japan (zip:350-8571) email akiyama@math.tsukuba.ac.jp
BENOÎT LORIDANT
Affiliation:
Lehrstuhl Mathematik und Statistik, Montanuniversität Leoben, Franz Josef Straße 18, A-8700 Leoben, Austria email loridant@geometrie.tuwien.ac.at
Corresponding
E-mail address:
akiyama@math.tsukuba.ac.jp
loridant@geometrie.tuwien.ac.at
Get access
Rights & Permissions[Opens in a new window]

Abstract

As an application of the boundary parametrization developed in our previous papers, we propose a new method to deduce information on the connected components of the interior of tiles. This gives a systematic way to study the topology of a certain class of self-affine tiles. An example due to Bandt and Gelbrich is examined to prove the efficiency of the method.

MSC classification

Primary: 28A80: Fractals 52C20: Tilings in $2$ dimensions
Type
Article
Information
Nagoya Mathematical Journal , Volume 226 , June 2017 , pp. 127 - 164
Copyright
© 2016 by The Editorial Board of the Nagoya Mathematical Journal  

Access options

Get access to the full version of this content by using one of the access options below.
If you should have access and can't see this content please contact technical support.

References

Adler, R. L., Symbolic dynamics and Markov partitions , Bull. Amer. Math. Soc. (N.S.) 35(1) (1998), 1–56; MR 1477538 (98j:58081).CrossRefGoogle Scholar
Arnoux, P. and Ito, S., Pisot substitutions and Rauzy fractals , Bull. Belg. Math. Soc. Simon Stevin 8(2) (2001), 181–207; Journées Montoises d’Informatique Théorique (Marne-la-Vallée, 2000); MR 1838930 (2002j:37018).Google Scholar
Akiyama, S. and Loridant, B., Boundary parametrization of planar self-affine tiles with collinear digit set , Sci. China Math. 53(9) (2010), 2173–2194; MR 2718819.CrossRefGoogle Scholar
Akiyama, S. and Loridant, B., Boundary parametrization of self-affine tiles , J. Math. Soc. Japan 63(2) (2011), 525–579; MR 2793110 (2012g:37034).CrossRefGoogle Scholar
Alexandrov, P. S., Combinatorial topology. Vol. 1, 2 and 3, Dover Publications, Mineola, NY, 1998, translated from the Russian, reprint of the 1956, 1957 and 1960 translations; MR 1643155 (99g:55001).Google Scholar
Alster, E., The finite number of interior component shapes of the Levy dragon , Discrete Comput. Geom. 43(4) (2010), 855–875; MR 2610474 (2011h:28009).CrossRefGoogle Scholar
Akiyama, S. and Sadahiro, T., A self-similar tiling generated by the minimal Pisot number , Acta Math. Info. Univ. Ostraviensis 6 (1998), 9–26.Google Scholar
Adler, R. L. and Weiss, B., Similarity of Automorphisms of the Torus, Memoirs of the American Mathematical Society 98 , American Mathematical Society, Providence, 1970, MR 0257315 (41 #1966).Google Scholar
Bandt, C. and Gelbrich, G., Classification of self-affine lattice tilings , J. Lond. Math. Soc. (2) 50(3) (1994), 581–593; MR 1299459 (95g:52035).CrossRefGoogle Scholar
Bailey, S., Kim, T. and Strichartz, R. S., Inside the Lévy dragon , Amer. Math. Monthly 109(8) (2002), 689–703; MR 1927621 (2003f:28011).Google Scholar
Bernat, J., Loridant, B. and Thuswaldner, J., Interior components of a tile associated to a quadratic canonical number system – Part II , Fractals 18(3) (2010), 385–397; MR 2683938.CrossRefGoogle Scholar
Baake, M. and Moody, R. V., Weighted Dirac combs with pure point diffraction , J. Reine Angew. Math. 573 (2004), 61–94; MR 2084582 (2006i:43009).Google Scholar
Bowen, R., Markov partitions for Axiom A diffeomorphisms , Amer. J. Math. 92 (1970), 725–747; MR 0277003 (43 #2740).CrossRefGoogle Scholar
Bowen, R., Markov partitions are not smooth , Proc. Amer. Math. Soc. 71(1) (1978), 130–132; MR 0474415 (57 #14055).CrossRefGoogle Scholar
Bandt, C. and Wang, Y., Disk-like self-affine tiles in ℝ2 , Discrete Comput. Geom. 26(4) (2001), 591–601; MR 1863811 (2002h:52028).CrossRefGoogle Scholar
Curry, E., Radix representations, self-affine tiles, and multivariable wavelets , Proc. Amer. Math. Soc. 134(8) (2006), 2411–2418; (electronic). MR 2213715 (2007m:52026).CrossRefGoogle Scholar
Dekking, M., Recurrent sets , Adv. in Math. 44(1) (1982), 78–104; MR 654549 (84e:52023).CrossRefGoogle Scholar
Dekking, M., Replicating superfigures and endomorphisms of free groups , J. Combin. Theory Ser. A 32(3) (1982), 315–320; MR 657046 (84i:51019).CrossRefGoogle Scholar
Duvall, P., Keesling, J. and Vince, A., The Hausdorff dimension of the boundary of a self-similar tile , J. London Math. Soc. (2) 61(3) (2000), 748–760; MR 1766102 (2001f:52045).CrossRefGoogle Scholar
Duan, S.-J., Liu, D. and Tang, T.-M., A planar integral self-affine tile with Cantor set intersections with its neighbors , Integers 9(A21) (2009), 227–237; MR 2534911 (2010h:52035).CrossRefGoogle Scholar
Dumont, J.-M. and Thomas, A., Systemes de numeration et fonctions fractales relatifs aux substitutions , Theoret. Comput. Sci. 65(2) (1989), 153–169; MR 1020484 (90m:11022).CrossRefGoogle Scholar
Dube, S., Undecidable problems in fractal geometry , Complex Systems 7(6) (1993), 423–444; MR 1307741 (96b:68042).Google Scholar
Gelbrich, G., Crystallographic reptiles , Geom. Dedicata 51(3) (1994), 235–256; MR 1293800 (95e:52041).CrossRefGoogle Scholar
Gröchenig, K. and Haas, A., Self-similar lattice tilings , J. Fourier Anal. Appl. 1(2) (1994), 131–170; MR 1348740 (96j:52037).CrossRefGoogle Scholar
Gilbert, W. J., Radix representations of quadratic fields , J. Math. Anal. Appl. 83(1) (1981), 264–274; MR 632342 (83m:12005).CrossRefGoogle Scholar
Gröchenig, K. and Madych, W. R., Multiresolution analysis, Haar bases, and self-similar tilings of R n , IEEE Trans. Inform. Theory 38(2, part 2) (1992), 556–568; MR 1162214 (93i:42001).CrossRefGoogle Scholar
Hutchinson, J. E., Fractals and self-similarity , Indiana Univ. Math. J. 30(5) (1981), 713–747; MR 625600 (82h:49026).CrossRefGoogle Scholar
Kenyon, R., “ Self-replicating tilings ”, in Symbolic dynamics and its applications (New Haven, CT, 1991), Contemp. Math. 135 , American Mathematical Society, Providence, RI, 1992, 239–263. MR 1185093 (94a:52043).CrossRefGoogle Scholar
Kenyon, R., The construction of self-similar tilings , Geom. Funct. Anal. 6(3) (1996), 471–488; MR 1392326 (97j:52025).CrossRefGoogle Scholar
Kellendonk, J., Lenz, D. and Savinien, J., Mathematics of Aperiodic Order, Progress in Mathematics 309 , Birkhäuser, Basel, 2015.Google Scholar
Loridant, B. and Luo, J., On a theorem of Bandt and Wang and its extension to p2-tiles , Discrete Comput. Geom. 41(4) (2009), 616–642; MR 2496320 (2010d:52044).CrossRefGoogle Scholar
Loridant, B., Topological properties of a class of cubic Rauzy fractals , Osaka J. Math. 53(1) (2016), 161–219; MR 3466830.Google Scholar
Loridant, B. and Thuswaldner, J. M., Interior components of a tile associated to a quadratic canonical number system , Topology Appl. 155(7) (2008), 667–695.CrossRefGoogle Scholar
Lagarias, J. C. and Wang, Y., Integral self-affine tiles in ℝ n . I. Standard and nonstandard digit sets , J. Lond. Math. Soc. (2) 54(1) (1996), 161–179; MR 1395075 (97f:52031).CrossRefGoogle Scholar
Lagarias, J. C. and Wang, Y., Self-affine tiles in R n , Adv. Math. 121(1) (1996), 21–49; MR 1399601 (97d:52034).CrossRefGoogle Scholar
Lagarias, J. C. and Wang, Y., Integral self-affine tiles in R n . II. Lattice tilings , J. Fourier Anal. Appl. 3(1) (1997), 83–102; MR 1428817 (98b:52026).CrossRefGoogle Scholar
Mauldin, R. D. and Williams, S. C., Hausdorff dimension in graph directed constructions , Trans. Amer. Math. Soc. 309(2) (1988), 811–829; MR 961615 (89i:28003).CrossRefGoogle Scholar
Ngai, S.-M. and Nguyen, N., The Heighway dragon revisited , Discrete Comput. Geom. 29(4) (2003), 603–623; MR 1976609 (2004d:28023).CrossRefGoogle Scholar
Ngai, S.-M. and Tang, T.-M., A technique in the topology of connected self-similar tiles , Fractals 12(4) (2004), 389–403; MR 2109984 (2006b:52018).CrossRefGoogle Scholar
Ngai, S.-M. and Tang, T.-M., Topology of connected self-similar tiles in the plane with disconnected interiors , Topology Appl. 150(1–3) (2005), 139–155; MR 2133675 (2006b:52019).CrossRefGoogle Scholar
Praggastis, B. L., Markov partitions for hyperbolic toral automorphisms, Ph.D. thesis, University of Washington, ProQuest LLC, Ann Arbor, MI, 1992;MR 2687824.Google Scholar
Rauzy, G., Nombres algébriques et substitutions , Bull. Soc. Math. France 110(2) (1982), 147–178; MR 667748 (84h:10074).CrossRefGoogle Scholar
Sinaĭ, Ja. G., Markov partitions and U-diffeomorphisms , Funktsional. Anal. i Prilozhen. 2(1) (1968), 64–89; MR 0233038 (38 #1361).CrossRefGoogle Scholar
Socolar, J., Tiling encyclopedia, http://tilings.math.uni-bielefeld.de/substitution/limhex.Google Scholar
Solomyak, B., Dynamics of self-similar tilings , Ergodic Theory Dynam. Systems 17(3) (1997), 695–738; MR 1452190 (98f:52030).CrossRefGoogle Scholar
Scheicher, K. and Thuswaldner, J. M., “ Neighbours of self-affine tiles in lattice tilings ”, in Fractals (Graz, 2001), Trends Math., Birkhäuser, Basel, 2003, 241–262. MR 2091708 (2005f:37039).CrossRefGoogle Scholar
Strichartz, R. S., Wavelets and self-affine tilings , Constr. Approx. 9(2–3) (1993), 327–346; MR 1215776 (94f:42039).CrossRefGoogle Scholar
Thurston, W., Groups, tilings, and finite state automata, AMS Colloquium Lecture Notes, Geometry Computing Group, 1989.Google Scholar
Vince, A., “ Digit tiling of Euclidean space ”, in Directions in Mathematical Quasicrystals, CRM Monogr. Ser. 13 , American Mathematical Society, Providence, RI, 2000, 329–370. MR 1798999 (2002g:52025).Google Scholar
Wang, Y., Wavelets, tiling, and spectral sets , Duke Math. J. 114(1) (2002), 43–57; MR 1915035 (2003e:42057).CrossRefGoogle Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 5
Total number of PDF views: 35 *
View data table for this chart

* Views captured on Cambridge Core between 19th October 2016 - 26th February 2021. This data will be updated every 24 hours.

1 Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

BOUNDARY PARAMETRIZATION AND THE TOPOLOGY OF TILES
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

BOUNDARY PARAMETRIZATION AND THE TOPOLOGY OF TILES
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

BOUNDARY PARAMETRIZATION AND THE TOPOLOGY OF TILES
Available formats
×
×

Reply to: Submit a response

Your details

Conflicting interests

Do you have any conflicting interests? *