Some remarks on the general finite type condition

Abstract

In this paper, the author discusses the iterated function system of generalized finite type conditions. First, the author constructs an iterated function system of generalized finite type condition, which satisfies the case where an invariant set is a basic set if and only if it is a subset of (0, 1). Second, the author proves that, with respect to the nested index sequence \{{\mathrm{\Lambda }}_k{\}}_k\ge \text{0}, any iterated function system in {\mathbb{R}}^dcannot satisfy the generalized finite type condition when the exponents of contractive ratios are not commensurable. Finally, the author constructs a family of self-similar sets which satisfy the generalized finite type condition and computes the Hausdorff dimensions of them.

Graphical abstract

Keywords

Self-similar set / iterated function system / generalized finite type condition / Hausdorff dimension

References

[1] Das M. , Ngai, S.M.. Graph-directed iterated function systems with overlaps. Indiana Univ. Math. J. 2004; 53(1): 109–134

[2] EdgarG.A., Measure, Topology, and Fractal Geometry, New York: Springer-Verlag, 1990

[3] FalconerK.J., Fractal Geometry: Mathematical Foundations and Applications, Chichester: John Wiley & Sons, 1990

[4] FalconerK.J., Techniques in Fractal Geometry, Chichester: John Wiley & Sons, 1997

[5] He X.G., Lau, K.S. , Rao, H.. Self-affine sets and graph-directed systems. Constr. Approx. 2003; 19(3): 373–397

[6] Hutchinson J.E.. Fractals and self-similarity. Indiana Univ. Math. J. 1981; 30(5): 713–747

[7] Jin N. , Yau, S.S.T.. General finite type IFS and M-matrix. Comm. Anal. Geom. 2005; 13(4): 821–843

[8] Lalley S.P.. B-expansions with deleted digits for Pisot numbers. Trans. Amer. Math. Soc. 1997; 349(11): 4355–4365

[9] Lau K.S. , Ngai, S.M.. Multifractal measures and a weak separation condition. Adv. Math. 1999; 141(1): 45–96

[10] Lau K.S. , Ngai, S.M.. A generalized finite type condition for iterated function systems. Adv. Math. 2007; 208(2): 647–671

[11] Lau K.S., Ngai, S.M. , Wang, X.Y.. Separation conditions for conformal iterated function systems. Monatsh. Math. 2009; 156(4): 325–355

[12] Mauldin R.D. , Williams, S.C.. Hausdorff dimension in graph directed constructions. Trans. Amer. Math. Soc. 1988; 309(2): 811–829

[13] Moran P.A.P.. Additive functions of intervals and Hausdorff measure. Proc. Cambridge Philos. Soc. 1946; 42: 15–23

[14] Ngai S.M. , Wang, Y.. Hausdorff dimension of self-similar sets with overlaps. J. London Math. Soc. 2001; 63(2): 655–672

[15] Nguyen N.T.. Iterated function systems of finite type and the weak separation property. Proc. Amer. Math. Soc. 2002; 130(2): 483–487

[16] Rao H. , Wen, Z.Y.. A class of self-similar fractals with overlap structure. Adv. in Appl. Math. 1998; 20(1): 50–72

[17] WenZ.Y., Mathematical Foundations of Fractal Geometry. Shanghai: Shanghai Scientific and Technological Education Publishing House, 2000 (in Chinese)

[18] Zerner M.P.W.. Weak separation properties for self-similar sets. Proc. Amer. Math. Soc. 1996; 124(11): 3529–3539

RIGHTS & PERMISSIONS

Higher Education Press 2025