×

A gluing lemma for iterated function systems. (English) Zbl 1337.28016

Summary: We obtain a special version of a gluing lemma related to the theory of iterated function systems (IFS). As an application, we verify that a family of concrete \(n\)-dimensional self-affine tiles \(\{T_{n,r} : 0 \leq r < 3, n \geq 3\}\) are homeomorphic with the unit cube \([0,1]^{n}\). The tiles \(T_{n,r}\) are “nontrivial” in the sense that each of them is neither a self-affine polytope nor the product of an interval with an \((n-1)\)-dimensional self-affine tile.

MSC:

28A80 Fractals
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. E. Hutchinson , Ind. Univ. Math. J. 30 , 713 ( 1981 ) . genRefLink(16, ’rf1’, ’10.1512
[2] J. Lagarias and Y. Wang , Adv. Math. 121 , 21 ( 1996 ) . genRefLink(16, ’rf2’, ’10.1006
[3] Q.-R. Deng and K.-S. Lau, J. Math. Anal. Appl. 380(2), 493 (2011). genRefLink(16, ’rf3’, ’10.1016
[4] J. Luo, H. Rao and B. Tan, Fractals 10(2), 223 (2002). [Abstract] genRefLink(128, ’rf4’, ’000176721400008’);
[5] M. F. Barnsley , Fractals Everywhere , 2nd edn. ( Academic Press Professional , Boston, MA , 1993 ) .
[6] J. Luo, S. Akiyama and J. M. Thuswaldner, Math. Proc. Cambridge Philos. Soc. 137(2), 397 (2004). genRefLink(16, ’rf6’, ’10.1017
[7] T.-M. Tang, Acta Math. Hungar. 109(4), 295 (2005). genRefLink(16, ’rf7’, ’10.1007
[8] J. Luo, Topology Appl. 154(3), 614 (2007). genRefLink(16, ’rf8’, ’10.1016
[9] J. Luo and J. M. Thuswaldner, Ann. Inst. Fourier (Grenoble) 56(7), 2493 (2006). genRefLink(16, ’rf9’, ’10.5802
[10] T. Jolivet, B. Loridant and J. Luo, J. Fractal Geom. 1(4), 427 (2014). genRefLink(16, ’rf10’, ’10.4171
[11] S.-M. Ngai and T.-M. Tang, Fractals 12(4), 389 (2004). [Abstract] genRefLink(128, ’rf11’, ’000226636100005’);
[12] B. Loridant and J. M. Thuswaldner, Topology Appl. 155(7), 667 (2008). genRefLink(16, ’rf12’, ’10.1016
[13] J. Bernat, B. Loridant and J. M. Thuswaldner, Fractals 18(3), 385 (2010). [Abstract] genRefLink(128, ’rf13’, ’000284706500012’);
[14] S. Akiyama and J. M. Thuswaldner , Geom. Dedicata 109 , 89 ( 2004 ) . genRefLink(16, ’rf14’, ’10.1007
[15] J. R. Munkres , Topology , 2nd edn. ( Prentice-Hall , Upper Saddle River, NJ , 1999 ) .
[16] M. A. Armstrong , Basic Topology ( Springer Science + Business Media, Inc. , 1983 ) . genRefLink(16, ’rf16’, ’10.1007
[17] M. Brown , Bull. Amer. Math. Soc. 66 , 74 ( 1960 ) . genRefLink(16, ’rf17’, ’10.1090
[18] G. T. Whyburn and E. Duda , Dynamic Topology , Undergraduate Texts in Mathematics ( Springer-Verlag , New York , 1979 ) . genRefLink(16, ’rf18’, ’10.1007
[19] C. Bandt and Y. Wang , Discrete Comput. Geom. 26 , 591 ( 2001 ) . genRefLink(16, ’rf19’, ’10.1007
[20] G. R. Conner and J. M. Thuswaldner, Self-affine manifolds, Available at , http://arxiv.org/ pdf/1402.3000.pdf . · Zbl 1350.28009
[21] H. Sagan, Math. Intell. 15(4), 37 (1993). genRefLink(16, ’rf21’, ’10.1007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.