Real Analysis Exchange

Thomas Jordan

Abstract

In this paper we continue the work started by Broomhead, Montaldi and Sidorov investigating the Hausdorff dimension of fat Sierpiński gaskets. We obtain generic results where the contraction rate $\lambda$ is in a certain region.

Article information

Source
Real Anal. Exchange, Volume 31, Number 1 (2005), 97-110.

Dates
First available in Project Euclid: 5 June 2006

https://projecteuclid.org/euclid.rae/1149516803

Mathematical Reviews number (MathSciNet)
MR2218191

Zentralblatt MATH identifier
1105.28004

Subjects
Secondary: 37C45: Dimension theory of dynamical systems

Citation

Jordan, Thomas. Dimension of fat Sierpiński gaskets. Real Anal. Exchange 31 (2005), no. 1, 97--110. https://projecteuclid.org/euclid.rae/1149516803

References

• C. J. Bishop, Topics in real analysis: Hausdorff dimension and fractals, lectures available at www.math.sunsyb.edu/$\sim$bishop/classes/math639.S01/math639.html
• D. Broomhead, J. Montaldi and N. Sidorov, Golden Gaskets: Variations on the Sierpiński sieve, Nonlinearity, 17, no. 4 (2004), 1455–1480.
• Manev Das, Sze-Man Ngai, Graph-directed iterated function systems with overlaps, Indiana Univ. Math. J., 53, no. 1 (2004), 109–134.
• K. Falconer, Fractal Geometry; Mathematical foundations and applications, Wiley, Chichester, 1990.
• B. Hunt and V. Kaloshin, How projections affect the dimension spectrum of fractal measures, Nonlinearity, 10, no 5 (1997), 1031–1046.
• J. Hutchinson, Fractals and self-similarity, Indiana University Math. J., 30 (1981), 271–280.
• Steven P. Lalley, $\beta$-expansions with deleted digits for Pisot numbers $\beta$, Trans. Amer. Math. Soc., 349, no. 11 (1997), 4355–4365.
• J. M. Marstrand, The dimension of Cartesian product sets, Proc. Camb. Phil. Soc., 50 (1954), 198–202.
• Sze-Man Ngai, Yang Wang, Hausdorff dimension of self-similar sets with overlaps, J. London Math. Soc. (2), 63, no. 3 (2001), 655–672.
• Y. Peres and B. Solomyak, Self-similar measures and the intersection of Cantor sets, Trans. Amer. Math. Soc., 350 (1998), 4065–4087.
• M. Pollicott and K. Simon, The Hausdorff dimension of $\lambda$-expansions with deleted digits, Trans. Amer. Math. Soc., 347, no. 3 (1995), 967–983.
• K. Simon and B. Solomyak, On the dimension of self-similar sets, Fractals, 10 (2003), 59–65.
• K. Simon, B. Solomyak and M. Urbanski, Invariant measures for parabolic IFS with overlaps and random continued fractions, Trans. Amer. Math. Soc., 353 (2001), 5145–5164.
• B. Solomyak, On the random series $\sum_{i=1}^{\infty}\pm\lambda^i$ (an Erdös problem), Annals of Maths, 142 (1995), 611–625.