Dimension and measure of baker-like skew-products of $\boldsymbol{\beta}$-transformations

David Färm; Tomas Persson

doi:10.3934/dcds.2012.32.3525

Article Contents

2012, Volume 32, Issue 10: 3525-3537. Doi: 10.3934/dcds.2012.32.3525

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Dimension and measure of baker-like skew-products of $\boldsymbol{\beta}$-transformations

David Färm^1, and
Tomas Persson^1,

1.
Institute of Mathematics, Polish Academy of Sciences, ulica Śniadeckich 8, P.O. Box 21, 00-956 Warszawa

Received: February 28, 2011

Revised: January 31, 2012

Published: September 2012

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Abstract

We consider a generalisation of the baker's transformation, consisting of a skew-product of contractions and a $\beta$-transformation. The Hausdorff dimension and Lebesgue measure of the attractor is calculated for a set of parameters with positive measure. The proofs use a new transverality lemma similar to Solomyak's [12]. This transversality, which is applicable to the considered class of maps holds for a larger set of parameters than Solomyak's transversality.

Keywords:

Dimension theory,
$\beta$-transformation.

Mathematics Subject Classification: Primary: 37D50, 37C40, 37C45.

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References

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