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Bi-invariant sets and measures have integer Hausdorff dimension

Published online by Cambridge University Press:  02 April 2001

DAVID MEIRI  and
YUVAL PERES
DAVID MEIRI
Affiliation:
The Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel (e-mail: [email protected])
YUVAL PERES
Affiliation:
The Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel (e-mail: [email protected]) Present address: Department of statistics, 367 Evans Hall, University of California, Berkeley CA 94720-3860, USA. (e-mail: [email protected])
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Abstract

Let $A,B$ be two diagonal endomorphisms of the $d$-dimensional torus with corresponding eigenvalues relatively prime. We show that for any $A$-invariant ergodic measure $\mu$, there exists a projection onto a torus ${\mathbb T}^r$ of dimension $r\ge\dim\mu$, that maps $\mu$-almost every $B$-orbit to a uniformly distributed sequence in ${\mathbb T}^r$. As a corollary we obtain that the Hausdorff dimension of any bi-invariant measure, as well as any closed bi-invariant set, is an integer.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 19 , Issue 2 , April 1999 , pp. 523 - 534
Copyright
1999 Cambridge University Press

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