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The unique measure of maximal entropy for a compact rank one locally CAT(0) space
Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $ \beta $-transformation
Department of Mathematics, University of Science and Technology of China, Hefei, 230026, Anhui, China |
$ \beta $ |
$ C $ |
$ \{a_n\} $ |
$ \begin{equation*} \liminf\limits_{N\to \infty}\frac{1}{N}\sum\limits_{j = 1}^{N}d(f^{a_j}(x),f^{a_j}(y )) = 0 \text{ and } \limsup\limits_{N\to \infty}\frac{1}{N}\sum\limits_{j = 1}^{N}d(f^{a_j}(x),f^{a_j}(y ))>0 \end{equation*} $ |
$ x $ |
$ y $ |
$ C $ |
References:
[1] |
F. Balibrea and V. Jiménez López,
The measure of scrambled sets: a survey, Acta Univ. M. Belii Ser. Math., 7 (1999), 3-11.
|
[2] |
F. Blanchard, W. Huang and L. Snoha,
Topological size of scrambled sets, Colloq. Math., 110 (2008), 293-361.
doi: 10.4064/cm110-2-3. |
[3] |
H. Bruin and V. Jiménez López,
On the Lebesgue measure of Li-Yorke pairs for interval maps, Comm. Math. Phys., 299 (2010), 523-560.
doi: 10.1007/s00220-010-1085-9. |
[4] |
J.-C. Ban and B. Li,
The multifractal spectra for the recurrence rates of beta-transformations, J. Math. Anal. Appl., 420 (2014), 1662-1679.
doi: 10.1016/j.jmaa.2014.06.051. |
[5] |
K. Dajani and C. Kraaikamp, Ergodic Theory of Numbers, Carus Mathematical Monographs, 29. Mathematical Association of America, Washington, DC, 2002. |
[6] |
T. Downarowicz,
Positive topological entropy implies chaos DC2, Proc. Amer. Math. Soc., 142 (2014), 137-149.
doi: 10.1090/S0002-9939-2013-11717-X. |
[7] |
K. Falconer, Fractal Geometry, , Mathematical foundations and applications. Third edition. John Wiley & Sons, Ltd., Chichester, 2014. |
[8] |
C. Fang, W. Huang, Y. Yi and P. Zhang,
Dimensions of stable sets and scrambled sets in positive finite entropy systems, Ergodic Theory Dynam. Systems, 32 (2012), 599-628.
doi: 10.1017/S0143385710000982. |
[9] |
F. Garcia-Ramos and L. Jin,
Mean proximality and mean Li-Yorke chaos, Proc. Amer. Math. Soc., 145 (2017), 2959-2969.
doi: 10.1090/proc/13440. |
[10] |
F. Hofbauer,
$\beta$-Shifts have unique maximal measure, Monatsh. Math., 85 (1978), 189-198.
doi: 10.1007/BF01534862. |
[11] |
W. Huang, J. Li and X. Ye,
Stable sets and mean Li-Yorke chaos in positive entropy systems, J. Funct. Anal., 266 (2014), 3377-3394.
doi: 10.1016/j.jfa.2014.01.005. |
[12] |
T. Y. Li and James A. Yorke,
Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992.
doi: 10.1080/00029890.1975.11994008. |
[13] |
K.-S. Lau and L. Shu,
The spectrum of Poincaré recurrence, Ergodic Theory Dynam. Systems, 28 (2008), 1917-1943.
doi: 10.1017/S0143385707001095. |
[14] |
B. Li and J. Wu,
Beta-expansion and continued fraction expansion, J. Math. Anal. Appl., 339 (2008), 1322-1331.
doi: 10.1016/j.jmaa.2007.07.070. |
[15] |
B. Li and Y.-C. Chen,
Chaotic and topological properties of $ \beta $-transformations, J. Math. Anal. Appl., 383 (2011), 585-596.
doi: 10.1016/j.jmaa.2011.05.049. |
[16] |
W. Liu and B. Li,
Chaotic and topological properties of continued fractions, J. Number Theory, 174 (2017), 369-383.
doi: 10.1016/j.jnt.2016.10.019. |
[17] |
J. Li and Y. Qiao,
Mean Li-Yorke chaos along some good sequences, Monatsh. Math., 186 (2018), 153-173.
doi: 10.1007/s00605-017-1086-2. |
[18] |
W.-B. Liu, C. Huang, M.-H. Li and S. Wang, A construction of the scrambled set with full Hausdorff dimension for beta-transformations, Fractals, 26 (2018), 1850005, 10pp.
doi: 10.1142/S0218348X18500056. |
[19] |
B. H. P. de M. e Maia, An Equivalent System for Studying Periodic Points of the Beta-Transformation for a Pisot or a Salem Number, Thesis (Ph.D.)-Universidade Autonoma de Lisboa (Portugal). 2008. |
[20] |
W. Parry,
On the $ \beta $-expansions of real numbers, Acta Math. Acad. Sci. Hungar., 11 (1960), 401-416.
doi: 10.1007/BF02020954. |
[21] |
A. Rényi,
Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., 8 (1957), 477-493.
doi: 10.1007/BF02020331. |
[22] |
S. Ito and Y. Takahashi,
Markov subshifts and realization of $ \beta $-expansions, J. Math. Soc. Japan, 26 (1974), 33-55.
doi: 10.2969/jmsj/02610033. |
[23] |
B. Schweizer and J. Smítal,
Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc., 344 (1994), 737-754.
doi: 10.1090/S0002-9947-1994-1227094-X. |
[24] |
Y. Wang, E. Chen and X. Zhou,
Mean Li-Yorke chaos for random dynamical systems, J. Differential Equations, 267 (2019), 2239-2260.
doi: 10.1016/j.jde.2019.03.012. |
[25] |
J. C. Xiong,
Hausdorff dimension of a chaotic set of shift of a symbolic space, Sci. China Ser. A, 38 (1995), 696-708.
|
show all references
References:
[1] |
F. Balibrea and V. Jiménez López,
The measure of scrambled sets: a survey, Acta Univ. M. Belii Ser. Math., 7 (1999), 3-11.
|
[2] |
F. Blanchard, W. Huang and L. Snoha,
Topological size of scrambled sets, Colloq. Math., 110 (2008), 293-361.
doi: 10.4064/cm110-2-3. |
[3] |
H. Bruin and V. Jiménez López,
On the Lebesgue measure of Li-Yorke pairs for interval maps, Comm. Math. Phys., 299 (2010), 523-560.
doi: 10.1007/s00220-010-1085-9. |
[4] |
J.-C. Ban and B. Li,
The multifractal spectra for the recurrence rates of beta-transformations, J. Math. Anal. Appl., 420 (2014), 1662-1679.
doi: 10.1016/j.jmaa.2014.06.051. |
[5] |
K. Dajani and C. Kraaikamp, Ergodic Theory of Numbers, Carus Mathematical Monographs, 29. Mathematical Association of America, Washington, DC, 2002. |
[6] |
T. Downarowicz,
Positive topological entropy implies chaos DC2, Proc. Amer. Math. Soc., 142 (2014), 137-149.
doi: 10.1090/S0002-9939-2013-11717-X. |
[7] |
K. Falconer, Fractal Geometry, , Mathematical foundations and applications. Third edition. John Wiley & Sons, Ltd., Chichester, 2014. |
[8] |
C. Fang, W. Huang, Y. Yi and P. Zhang,
Dimensions of stable sets and scrambled sets in positive finite entropy systems, Ergodic Theory Dynam. Systems, 32 (2012), 599-628.
doi: 10.1017/S0143385710000982. |
[9] |
F. Garcia-Ramos and L. Jin,
Mean proximality and mean Li-Yorke chaos, Proc. Amer. Math. Soc., 145 (2017), 2959-2969.
doi: 10.1090/proc/13440. |
[10] |
F. Hofbauer,
$\beta$-Shifts have unique maximal measure, Monatsh. Math., 85 (1978), 189-198.
doi: 10.1007/BF01534862. |
[11] |
W. Huang, J. Li and X. Ye,
Stable sets and mean Li-Yorke chaos in positive entropy systems, J. Funct. Anal., 266 (2014), 3377-3394.
doi: 10.1016/j.jfa.2014.01.005. |
[12] |
T. Y. Li and James A. Yorke,
Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992.
doi: 10.1080/00029890.1975.11994008. |
[13] |
K.-S. Lau and L. Shu,
The spectrum of Poincaré recurrence, Ergodic Theory Dynam. Systems, 28 (2008), 1917-1943.
doi: 10.1017/S0143385707001095. |
[14] |
B. Li and J. Wu,
Beta-expansion and continued fraction expansion, J. Math. Anal. Appl., 339 (2008), 1322-1331.
doi: 10.1016/j.jmaa.2007.07.070. |
[15] |
B. Li and Y.-C. Chen,
Chaotic and topological properties of $ \beta $-transformations, J. Math. Anal. Appl., 383 (2011), 585-596.
doi: 10.1016/j.jmaa.2011.05.049. |
[16] |
W. Liu and B. Li,
Chaotic and topological properties of continued fractions, J. Number Theory, 174 (2017), 369-383.
doi: 10.1016/j.jnt.2016.10.019. |
[17] |
J. Li and Y. Qiao,
Mean Li-Yorke chaos along some good sequences, Monatsh. Math., 186 (2018), 153-173.
doi: 10.1007/s00605-017-1086-2. |
[18] |
W.-B. Liu, C. Huang, M.-H. Li and S. Wang, A construction of the scrambled set with full Hausdorff dimension for beta-transformations, Fractals, 26 (2018), 1850005, 10pp.
doi: 10.1142/S0218348X18500056. |
[19] |
B. H. P. de M. e Maia, An Equivalent System for Studying Periodic Points of the Beta-Transformation for a Pisot or a Salem Number, Thesis (Ph.D.)-Universidade Autonoma de Lisboa (Portugal). 2008. |
[20] |
W. Parry,
On the $ \beta $-expansions of real numbers, Acta Math. Acad. Sci. Hungar., 11 (1960), 401-416.
doi: 10.1007/BF02020954. |
[21] |
A. Rényi,
Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., 8 (1957), 477-493.
doi: 10.1007/BF02020331. |
[22] |
S. Ito and Y. Takahashi,
Markov subshifts and realization of $ \beta $-expansions, J. Math. Soc. Japan, 26 (1974), 33-55.
doi: 10.2969/jmsj/02610033. |
[23] |
B. Schweizer and J. Smítal,
Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc., 344 (1994), 737-754.
doi: 10.1090/S0002-9947-1994-1227094-X. |
[24] |
Y. Wang, E. Chen and X. Zhou,
Mean Li-Yorke chaos for random dynamical systems, J. Differential Equations, 267 (2019), 2239-2260.
doi: 10.1016/j.jde.2019.03.012. |
[25] |
J. C. Xiong,
Hausdorff dimension of a chaotic set of shift of a symbolic space, Sci. China Ser. A, 38 (1995), 696-708.
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