Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-23T11:33:42.640Z Has data issue: false hasContentIssue false
  • English
  • Français

Hausdorff dimension for fractals invariant under multiplicative integers

Published online by Cambridge University Press:  23 November 2011

RICHARD KENYON ,
YUVAL PERES  and
BORIS SOLOMYAK
RICHARD KENYON
Affiliation:
Department of Mathematics, Brown University, Providence, RI 02912, USA
YUVAL PERES
Affiliation:
Microsoft Research, 1 Microsoft Way, Redmond, WA 98052, USA
BORIS SOLOMYAK
Affiliation:
Department of Mathematics, University of Washington, Box 354350, Seattle WA 98195, USA (email: solomyak@math.washington.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider subsets of the (symbolic) sequence space that are invariant under the action of the semigroup of multiplicative integers. A representative example is the collection of all 0–1 sequences (xk) such that xkx2k=0 for all k. We compute the Hausdorff and Minkowski dimensions of these sets and show that they are typically different. The proof proceeds via a variational principle for multiplicative subshifts.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 5 , October 2012 , pp. 1567 - 1584
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bedford, T.. Crinkly curves, Markov partitions and box dimension in self-similar sets. PhD Thesis, University of Warwick, 1984.Google Scholar
[2]Billingsley, P.. Ergodic Theory and Information. Wiley, New York, 1965.Google Scholar
[3]Dajani, K. and Kraaikamp, C.. Ergodic Theory of Numbers (Carus Mathematical Monographs, 29). Mathematical Association of America, Washington, DC, 2002.Google Scholar
[4]Falconer, K.. Techniques in Fractal Geometry. John Wiley & Sons, Chichester, 1997.Google Scholar
[5]Fan, A., Liao, L. and Ma, J.. Level sets of multiple ergodic averages. arXiv:1105.3032.Google Scholar
[6]Fan, A., Schmeling, J. and Wu, M.. Multifractal analysis of multiple ergodic averages. Preprint, 2011, arXiv:1108.4131.Google Scholar
[7]Furstenberg, H.. Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Syst. Theory 1 (1967), 1–49.Google Scholar
[8]Kenyon, R., Peres, Y. and Solomyak, B.. Hausdorff dimension of the multiplicative golden mean shift. C. R. Acad. Sci. Paris, Ser. I 349 (2011), 625–628.CrossRefGoogle Scholar
[9]Lyons, R.. Random walks and percolation on trees. Ann. Probab. 18 (1990), 931–958.CrossRefGoogle Scholar
[10]McMullen, C.. The Hausdorff dimension of general Sierpinski carpets. Nagoya Math. J. 96 (1984), 1–9.CrossRefGoogle Scholar
[11]Parry, W.. On the β-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416.Google Scholar
[12]Peres, Y. and Solomyak, B.. Dimension spectrum for a nonconventional ergodic average. Preprint, 2011, arXiv:1107.1749.Google Scholar