no. 11-12
Mathematical Analysis/Dynamical Systems
Hausdorff dimension of the multiplicative golden mean shift
[Dimension de Hausdorff du shift de Fibonacci multiplicatif]
Présenté par : Jean-Pierre Kahane
Kenyon, Richard1 ; Peres, Yuval2 ; Solomyak, Boris3
1 Department of Mathematics, Brown University, Box 1917, 151 Thayer Street, Providence, RI 02912, USA
2 Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA
3 Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195-4350, USA
Comptes Rendus. Mathématique, Tome 349 (2011) no. 11-12, pp. 625-628

We compute the Hausdorff dimension of the “multiplicative golden mean shift” defined as the set of all reals in [0,1] whose binary expansion (xk) satisfies xkx2k=0 for all k⩾1, and show that it is smaller than the Minkowski dimension.

Nous calculons la dimension de Hausdorff du « shift de Fibonacci multiplicatif », cʼest-à-dire lʼensemble des nombres réels dans [0,1] dont le développement en binaire (xk) satisfait xkx2k=0 pour tout k⩾1. Nous montrons que la dimension de Hausdorff est plus petite que la dimension de Minkowski.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.05.009

Kenyon, Richard 1 ; Peres, Yuval 2 ; Solomyak, Boris 3

1 Department of Mathematics, Brown University, Box 1917, 151 Thayer Street, Providence, RI 02912, USA
2 Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA
3 Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195-4350, USA 
@article{CRMATH_2011__349_11-12_625_0,
     author = {Kenyon, Richard and Peres, Yuval and Solomyak, Boris},
     title = {Hausdorff dimension of the multiplicative golden mean shift},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {625--628},
     year = {2011},
     publisher = {Elsevier},
     volume = {349},
     number = {11-12},
     doi = {10.1016/j.crma.2011.05.009},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.crma.2011.05.009/}
}
TY  - JOUR
AU  - Kenyon, Richard
AU  - Peres, Yuval
AU  - Solomyak, Boris
TI  - Hausdorff dimension of the multiplicative golden mean shift
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 625
EP  - 628
VL  - 349
IS  - 11-12
PB  - Elsevier
UR  - https://www.numdam.org/articles/10.1016/j.crma.2011.05.009/
DO  - 10.1016/j.crma.2011.05.009
LA  - en
ID  - CRMATH_2011__349_11-12_625_0
ER  - 
%0 Journal Article
%A Kenyon, Richard
%A Peres, Yuval
%A Solomyak, Boris
%T Hausdorff dimension of the multiplicative golden mean shift
%J Comptes Rendus. Mathématique
%D 2011
%P 625-628
%V 349
%N 11-12
%I Elsevier
%U https://www.numdam.org/articles/10.1016/j.crma.2011.05.009/
%R 10.1016/j.crma.2011.05.009
%G en
%F CRMATH_2011__349_11-12_625_0
Kenyon, Richard; Peres, Yuval; Solomyak, Boris. Hausdorff dimension of the multiplicative golden mean shift. Comptes Rendus. Mathématique, Tome 349 (2011) no. 11-12, pp. 625-628. doi: 10.1016/j.crma.2011.05.009

[1] T. Bedford, Crinkly curves, Markov partitions and box dimension in self-similar sets, PhD thesis, University of Warwick, 1984.

[2] Billingsley, P. Ergodic Theory and Information, Wiley, New York, 1965

[3] Falconer, K. Techniques in Fractal Geometry, John Wiley & Sons, Chichester, 1997

[4] Fan, A.; Liao, L.; Ma, J. Level sets of multiple ergodic averages (preprint) | arXiv

[5] Furstenberg, H. Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory, Volume 1 (1967), pp. 1-49

[6] Kenyon, R.; Peres, Y.; Solomyak, B. Hausdorff dimension for fractals invariant under the multiplicative integers, 2011 (preprint) | arXiv

[7] McMullen, C. The Hausdorff dimension of general Sierpinski carpets, Nagoya Math. J., Volume 96 (1984), pp. 1-9

Cité par Sources :