Abstract
An n ×n matrix M is called expanding if all its eigenvalues have moduli > 1. Let A be a nonempty finite set of vectors in the n-dimensional Euclidean space. Then there exists a unique nonempty compact set F satisfying \(MF\,=\,F + A\). F is called a self-affine set or a self-affine fractal. F can also be considered as the attractor of an affine iterated function system. Although such sets are basic structures in the theory of fractals, there are still many problems on them to be studied. Among those problems, the calculation or the estimation of fractal dimensions of F is of considerable interest. In this work, we discuss some problems about the singular value dimension of self-affine sets. We then generalize the singular value dimension to certain graph directed sets and give a result on the computation of it. On the other hand, for a very few classes of self-affine fractals, the Hausdorff dimension and the singular value dimension are known to be different. Such fractals are called exceptional self-affine fractals. Finally, we present a new class of exceptional self-affine fractals and show that the generalized singular value dimension of F in that class is the same as the box (counting) dimension.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bedford, T., Urbański, M.: The box and Hausdorff dimension of self-affine sets. Ergod. Theor. Dyn. Syst. 10, 627–644 (1990)
Falconer, K.J.: The Hausdorff dimension of self-affine fractals. Math. Proc. Camb. Philos. Soc. 103, 339–350 (1988)
Falconer, K.J.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, Chichester (1990)
Falconer, K.J.: The dimension of self-affine fractals II. Math. Proc. Camb. Philos. Soc. 111, 169–179 (1992)
Falconer, K.J., Miao, J.: Dimensions of self-affine fractals and multifractals generated by upper triangular matrices. Fractals 15(3), 289–299 (2007)
He, X.-G., Lau, K.-S.: On a generalized dimension of self-affine fractals. Math. Nachr. 281(8), 1142–1158 (2008)
He, X.-G., Lau, K.-S., Rao, H.: Self-affine sets and graph-directed systems. Constr. Approx. 19(3), 373–397 (2003)
Hueter, I., Lalley, S.P.: Falconer’s formula for the Hausdorff dimension of a self-affine set in \({\mathbb{R}}^{2}\). Ergod. Theor. Dyn. Syst. 15, 77–97 (1995)
Kirat, I.: Disk-like tiles and self-affine curves with non-collinear digits. Math. Comp. 79(6), 1019–1045 (2010)
Kirat, I., Kocyigit, I.: A new class of exceptional self-affine fractals. J. Math. Anal. Appl. (2012) doi:10.1016/j.jmaa.2012.10.065 (in press)
Lagarias, J.C., Wang, Y.: Integral self-affine tiles in \({\mathbb{R}}^{n}\). Adv. Math. 121, 21–49 (1996)
Mauldin, R.D., Williams, S.C.: Hausdorff dimension in graph directed constructions. Trans. Am. Math. Soc. 309, 811–829 (1988)
McMullen, C.: The Hausdorff dimension of general Sierpinski carpets. Nagoya Math. J. 96, 1–9 (1984)
Paulsen, W.-H.: Lower bounds for the Hausdorff dimension of n-dimensional self-affine sets. Chaos, Solitons and Fractals 5(6), 909–931 (1995)
Simon, K., Solomyak, B.: On the dimension of self-similar sets. Fractals 10(1), 59–66 (2002)
Solomyak, B.: Measure and dimension for some fractal families. Math. Proc. Camb. Philos. Soc. 124, 531–546 (1998)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kirat, I., Kocyigit, I. (2013). On the Dimension of Self-Affine Fractals. In: Stavrinides, S., Banerjee, S., Caglar, S., Ozer, M. (eds) Chaos and Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33914-1_19
Download citation
DOI: https://doi.org/10.1007/978-3-642-33914-1_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-33913-4
Online ISBN: 978-3-642-33914-1
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)