Comptes Rendus
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

Richard Kenyon1 ; Yuval Peres2 ; Boris Solomyak3
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, Volume 349 (2011) no. 11-12, pp. 625-628.

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.

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.

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DOI : 10.1016/j.crma.2011.05.009

Richard Kenyon 1 ; Yuval Peres 2 ; Boris Solomyak 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 
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Richard Kenyon; Yuval Peres; Boris Solomyak. Hausdorff dimension of the multiplicative golden mean shift. Comptes Rendus. Mathématique, Volume 349 (2011) no. 11-12, pp. 625-628. doi : 10.1016/j.crma.2011.05.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.05.009/

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

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

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

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

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

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

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