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Formation of singularities to quasi-linear hyperbolic systems with initial data of small BV norm
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 |
References:
[1] |
J. C. Alexander and J. A. Yorke, Fat baker's transformations,, Ergodic Theory & Dynamical Systems, 4 (1984), 1.
|
[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. |
[3] |
P. Erdős, On a family of symmetric Bernoulli convolutions,, American Journal of Mathematics, 61 (1939), 974.
doi: 10.2307/2371641. |
[4] |
K. Falconer, "Fractal Geometry. Mathematical Foundations and Applications,", Second edition, (2003).
|
[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. |
[6] |
W. Parry, On the $\beta$-expansion of real numbers,, Acta Mathematica Academiae Scientiarum Hungaricae, 11 (1960), 401.
doi: 10.1007/BF02020954. |
[7] |
Y. Peres and B. Solomyak, Absolute continuity of Bernoulli convolutions, a simple proof,, Mathematical Research Letters, 3 (1996), 231.
|
[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. |
[9] |
A. Rényi, Representations for real numbers and their ergodic properties,, Acta Mathematica Academiae Scientiarum Hungaricae, 8 (1957), 477.
|
[10] |
E. Sataev, Ergodic properties of the Belykh map,, Journal of Mathematical Sciences, 95 (1999), 2564.
doi: 10.1007/BF02169056. |
[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. |
[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. |
show all references
References:
[1] |
J. C. Alexander and J. A. Yorke, Fat baker's transformations,, Ergodic Theory & Dynamical Systems, 4 (1984), 1.
|
[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. |
[3] |
P. Erdős, On a family of symmetric Bernoulli convolutions,, American Journal of Mathematics, 61 (1939), 974.
doi: 10.2307/2371641. |
[4] |
K. Falconer, "Fractal Geometry. Mathematical Foundations and Applications,", Second edition, (2003).
|
[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. |
[6] |
W. Parry, On the $\beta$-expansion of real numbers,, Acta Mathematica Academiae Scientiarum Hungaricae, 11 (1960), 401.
doi: 10.1007/BF02020954. |
[7] |
Y. Peres and B. Solomyak, Absolute continuity of Bernoulli convolutions, a simple proof,, Mathematical Research Letters, 3 (1996), 231.
|
[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. |
[9] |
A. Rényi, Representations for real numbers and their ergodic properties,, Acta Mathematica Academiae Scientiarum Hungaricae, 8 (1957), 477.
|
[10] |
E. Sataev, Ergodic properties of the Belykh map,, Journal of Mathematical Sciences, 95 (1999), 2564.
doi: 10.1007/BF02169056. |
[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. |
[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. |
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