Hausdorff dimension of the multiplicative golden mean shift

Richard Kenyon; Yuval Peres; Boris Solomyak

doi:10.1016/j.crma.2011.05.009

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 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

Comptes Rendus. Mathématique, Volume 349 (2011) no. 11-12, pp. 625-628.

Résumé
Abstract

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 $(x_{k})$ satisfait $x_{k} x_{2 k} = 0$ 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 $(x_{k})$ satisfies $x_{k} x_{2 k} = 0$ for all $k ⩾ 1$ , and show that it is smaller than the Minkowski dimension.

Reçu le : 2011-05-02
Accepté le : 2011-05-17
Publié le : 2011-06-01

DOI : 10.1016/j.crma.2011.05.009

Affiliations des auteurs :

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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     author = {Richard Kenyon and Yuval Peres and Boris Solomyak},
     title = {Hausdorff dimension of the multiplicative golden mean shift},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {625--628},
     publisher = {Elsevier},
     volume = {349},
     number = {11-12},
     year = {2011},
     doi = {10.1016/j.crma.2011.05.009},
     language = {en},
}

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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/

Bibliographie
Cité par

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