Abstract
Let \(E\) be the self-similar set generated by the iterated function system
\(f_0(x)=\frac{x}{\beta},\quad f_1(x)=\frac{x+1}{\beta}, \quad f_{\beta+1}=\frac{x+\beta+1}{\beta}\)
with \(\beta\ge 3\). Then \(E\) is a self-similar set with complete overlaps, i.e., \(f_{0}\circ f_{\beta+1}={f_{1}\circ f_1}\), but \(E\) is not totally self-similar. We investigate all of its generating iterated function systems, give the spectrum of \(E\), and determine the Hausdorff dimensions and Hausdorff measures of \(E\) and of the sets which contain all points in \(E\) having finite or infinite different codings.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Akiyama, S., Komornik, V.: Discrete spectra and Pisot numbers. J. Number Theory 133, 375–390 (2013)
Broomhead, D., Montaldi, J., Sidorov, N.: Golden gaskets: variations on the Sierpiński sieve. Nonlinearity 17, 1455–1480 (2004)
Dajani, K., Jiang, K., Kong, D., Li, W.: Multiple expansions of real numbers with digits set \(\{0,1, q\}\). Math. Z. 291, 1605–1619 (2019)
K. Dajani, K. Jiang, D. Kong, W. Li and L. Xi, Multiple codings for self-similar sets with overlaps, arXiv:1603.09304
Dajani, K., Kong, D., Yao, Y.: On the structure of \(\lambda \)-Cantor set with overlaps. Adv. Appl. Math. 108, 97–125 (2019)
Deng, Q., Lau, K.-S.: On the equivalence of homogeneous iterated function systems. Nonlinearity 26, 2767–2775 (2013)
Deng, Q., Lau, K.-S.: Structure of the class of iterated function systems that generate the same self-similar set. J. Fractal Geom. 4, 43–71 (2017)
Erdős, P., Komornik, V.: Developments in non-integer bases. Acta Math. Hungar. 79, 57–83 (1998)
K. J. Falconer, Fractal Geometry. Mathematical Foundations and Applications, 3rd ed., John Wiley & Sons (Chichester, 2014)
Feng, D.-J.: On the topology of polynomials with bounded integer coefficients. J. Eur. Math. Soc. 18, 181–193 (2016)
Feng, D.-J., Wang, Y.: On the structures of generating iterated function systems of Cantor sets. Adv. Math. 222, 1964–1981 (2009)
Feng, D.-J., Wang, Y.: Bernoulli convolutions associated with certain non-Pisot numbers. Adv. Math. 187, 173–194 (2004)
Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. 30, 713–747 (1981)
V. Komornik, Expansions in noninteger bases, Integers, 11B (2011), Paper No. A9, 30 pp
D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press (Cambridge, 1995)
Mauldin, D., Williams, C.: Hausdorff dimension in graph directed constructions. Trans. Amer. Math. Soc. 309, 811–829 (1988)
Ngai, S.-M., Wang, Y.: Hausdorff dimension of self-similar sets with overlaps. J. London Math. Soc. 63, 655–672 (2001)
Peres, Y., Solomyak, B.: Approximation by polynomials with coefficients \(\pm 1\). J. Number Theory 84, 185–198 (2000)
N. Sidorov, Arithmetic dynamics, in: Topics in Dynamics and Ergodic Theory, London Math. Soc. Lecture Note Ser., vol. 310, Cambridge University Press (Cambridge, 2003), pp. 145–189
Yao, Y., Li, W.: Generating iterated function systems for a class of self-similar sets with complete overlap. Publ. Math. Debrecen 87, 23–33 (2015)
Yao, Y.: Generating iterated function systems for self-similar sets with a separation condition. Fund. Math. 237, 127–133 (2017)
Zou, Y., Lu, J., Li, W.: Unique expansion of points of a class of self-similar sets with overlaps. Mathematika 58, 371–388 (2012)
Acknowledgement
The authors thank the anonymous referee for many useful remarks.
Author information
Authors and Affiliations
Corresponding author
Additional information
D. Kong was supported by NSFC No. 11971079 and the Fundamental and Frontier Research Project of Chongqing No. cstc2019jcyj-msxmX0338.
Rights and permissions
About this article
Cite this article
Kong, D., Yao, Y. On a kind of self-similar sets with complete overlaps. Acta Math. Hungar. 163, 601–622 (2021). https://doi.org/10.1007/s10474-020-01116-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-020-01116-4