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October  2012, 32(10): 3525-3537. doi: 10.3934/dcds.2012.32.3525

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, Poland, Poland

Received  March 2011 Revised  February 2012 Published  May 2012

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: $\beta$-transformation., Dimension theory.
Mathematics Subject Classification: Primary: 37D50, 37C40, 37C4.
Citation: David Färm, Tomas Persson. Dimension and measure of baker-like skew-products of $\boldsymbol{\beta}$-transformations. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3525-3537. doi: 10.3934/dcds.2012.32.3525
References:
[1]

J. C. Alexander and J. A. Yorke, Fat baker's transformations, Ergodic Theory & Dynamical Systems, 4 (1984), 1-23.  Google Scholar

[2]

G. Brown and Q. Yin, $\beta$-transformation, natural extension and invariant measure, Ergodic Theory and Dynamical Systems, 20 (2000), 1271-1285. doi: 10.1017/S0143385700000699.  Google Scholar

[3]

P. Erdős, On a family of symmetric Bernoulli convolutions, American Journal of Mathematics, 61 (1939), 974-976. doi: 10.2307/2371641.  Google Scholar

[4]

K. Falconer, "Fractal Geometry. Mathematical Foundations and Applications," Second edition, John Wiley & Sons, Inc., Hoboken, NJ, 2003.  Google Scholar

[5]

D. Kwon, The natural extensions of $\beta$-transformations which generalize baker's transformations, Nonlinearity, 22 (2009), 301-310. doi: 10.1088/0951-7715/22/2/004.  Google Scholar

[6]

W. Parry, On the $\beta$-expansion of real numbers, Acta Mathematica Academiae Scientiarum Hungaricae, 11 (1960), 401-416. doi: 10.1007/BF02020954.  Google Scholar

[7]

Y. Peres and B. Solomyak, Absolute continuity of Bernoulli convolutions, a simple proof, Mathematical Research Letters, 3 (1996), 231-239.  Google Scholar

[8]

Ya. Pesin, Dynamical systems with generalized hyperbolic attractors: Hyperbolic, ergodic and topological properties, Ergodic Theory and Dynamical Systems, 12 (1992), 123-151. doi: 10.1017/S0143385700006635.  Google Scholar

[9]

A. Rényi, Representations for real numbers and their ergodic properties, Acta Mathematica Academiae Scientiarum Hungaricae, 8 (1957), 477-493.  Google Scholar

[10]

E. Sataev, Ergodic properties of the Belykh map, Journal of Mathematical Sciences, 95 (1999), 2564-2575. doi: 10.1007/BF02169056.  Google Scholar

[11]

J. Schmeling and S. Troubetzkoy, Dimension and invertibility of hyperbolic endomorphisms with singularities, Ergodic Theory and Dynamical Systems, 18 (1998), 1257-1282. doi: 10.1017/S0143385798117996.  Google Scholar

[12]

B. Solomyak, On the random series $\sum \pm \lambda^n$ (an Erdős problem), Annals of Mathematics (2), 142 (1995), 611-625. doi: 10.2307/2118556.  Google Scholar

show all references

References:
[1]

J. C. Alexander and J. A. Yorke, Fat baker's transformations, Ergodic Theory & Dynamical Systems, 4 (1984), 1-23.  Google Scholar

[2]

G. Brown and Q. Yin, $\beta$-transformation, natural extension and invariant measure, Ergodic Theory and Dynamical Systems, 20 (2000), 1271-1285. doi: 10.1017/S0143385700000699.  Google Scholar

[3]

P. Erdős, On a family of symmetric Bernoulli convolutions, American Journal of Mathematics, 61 (1939), 974-976. doi: 10.2307/2371641.  Google Scholar

[4]

K. Falconer, "Fractal Geometry. Mathematical Foundations and Applications," Second edition, John Wiley & Sons, Inc., Hoboken, NJ, 2003.  Google Scholar

[5]

D. Kwon, The natural extensions of $\beta$-transformations which generalize baker's transformations, Nonlinearity, 22 (2009), 301-310. doi: 10.1088/0951-7715/22/2/004.  Google Scholar

[6]

W. Parry, On the $\beta$-expansion of real numbers, Acta Mathematica Academiae Scientiarum Hungaricae, 11 (1960), 401-416. doi: 10.1007/BF02020954.  Google Scholar

[7]

Y. Peres and B. Solomyak, Absolute continuity of Bernoulli convolutions, a simple proof, Mathematical Research Letters, 3 (1996), 231-239.  Google Scholar

[8]

Ya. Pesin, Dynamical systems with generalized hyperbolic attractors: Hyperbolic, ergodic and topological properties, Ergodic Theory and Dynamical Systems, 12 (1992), 123-151. doi: 10.1017/S0143385700006635.  Google Scholar

[9]

A. Rényi, Representations for real numbers and their ergodic properties, Acta Mathematica Academiae Scientiarum Hungaricae, 8 (1957), 477-493.  Google Scholar

[10]

E. Sataev, Ergodic properties of the Belykh map, Journal of Mathematical Sciences, 95 (1999), 2564-2575. doi: 10.1007/BF02169056.  Google Scholar

[11]

J. Schmeling and S. Troubetzkoy, Dimension and invertibility of hyperbolic endomorphisms with singularities, Ergodic Theory and Dynamical Systems, 18 (1998), 1257-1282. doi: 10.1017/S0143385798117996.  Google Scholar

[12]

B. Solomyak, On the random series $\sum \pm \lambda^n$ (an Erdős problem), Annals of Mathematics (2), 142 (1995), 611-625. doi: 10.2307/2118556.  Google Scholar

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