# American Institute of Mathematical Sciences

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

 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.
Citation: David Färm, Tomas Persson. Dimension and measure of baker-like skew-products of $\boldsymbol{\beta}$-transformations. Discrete & Continuous Dynamical Systems - A, 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.   Google Scholar [2] G. Brown and Q. Yin, $\beta$-transformation, natural extension and invariant measure,, Ergodic Theory and Dynamical Systems, 20 (2000), 1271.  doi: 10.1017/S0143385700000699.  Google Scholar [3] P. Erdős, On a family of symmetric Bernoulli convolutions,, American Journal of Mathematics, 61 (1939), 974.  doi: 10.2307/2371641.  Google Scholar [4] K. Falconer, "Fractal Geometry. Mathematical Foundations and Applications,", Second edition, (2003).   Google Scholar [5] D. Kwon, The natural extensions of $\beta$-transformations which generalize baker's transformations,, Nonlinearity, 22 (2009), 301.  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.  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.   Google Scholar [8] Ya. Pesin, Dynamical systems with generalized hyperbolic attractors: Hyperbolic, ergodic and topological properties,, Ergodic Theory and Dynamical Systems, 12 (1992), 123.  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.   Google Scholar [10] E. Sataev, Ergodic properties of the Belykh map,, Journal of Mathematical Sciences, 95 (1999), 2564.  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.  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.  doi: 10.2307/2118556.  Google Scholar

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##### References:
 [1] J. C. Alexander and J. A. Yorke, Fat baker's transformations,, Ergodic Theory & Dynamical Systems, 4 (1984), 1.   Google Scholar [2] G. Brown and Q. Yin, $\beta$-transformation, natural extension and invariant measure,, Ergodic Theory and Dynamical Systems, 20 (2000), 1271.  doi: 10.1017/S0143385700000699.  Google Scholar [3] P. Erdős, On a family of symmetric Bernoulli convolutions,, American Journal of Mathematics, 61 (1939), 974.  doi: 10.2307/2371641.  Google Scholar [4] K. Falconer, "Fractal Geometry. Mathematical Foundations and Applications,", Second edition, (2003).   Google Scholar [5] D. Kwon, The natural extensions of $\beta$-transformations which generalize baker's transformations,, Nonlinearity, 22 (2009), 301.  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.  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.   Google Scholar [8] Ya. Pesin, Dynamical systems with generalized hyperbolic attractors: Hyperbolic, ergodic and topological properties,, Ergodic Theory and Dynamical Systems, 12 (1992), 123.  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.   Google Scholar [10] E. Sataev, Ergodic properties of the Belykh map,, Journal of Mathematical Sciences, 95 (1999), 2564.  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.  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.  doi: 10.2307/2118556.  Google Scholar
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