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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: [email protected])
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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

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