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February  2021, 41(2): 525-536. doi: 10.3934/dcds.2020267

Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $ \beta $-transformation

Yuanfen Xiao

Department of Mathematics, University of Science and Technology of China, Hefei, 230026, Anhui, China

Received  December 2019 Revised  April 2020 Published  July 2020

Fund Project: The author is supported by NNSF of China grant 11871228

We construct a mean Li-Yorke chaotic set along polynomial sequences (the degree of this polynomial is not less than three) with full Hausdorff dimension and full topological entropy for
$ \beta $
-transformation. An uncountable subset
$ C $
 is said to be a mean Li-Yorke chaotic set along sequence
$ \{a_n\} $
, if both
$ \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*} $
hold for any two distinct points
$ x $
 and
$ y $
 in
$ C $
.
Keywords: $ \beta $-transformation, mean Li-Yorke chaos, sequence version of chaos, Hausdorff dimension, topological entropy.
Mathematics Subject Classification: Primary:54H20, 37C45, 37D45;Secondary:37B40.
Citation: Yuanfen Xiao. Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $ \beta $-transformation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 525-536. doi: 10.3934/dcds.2020267
References:
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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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W. Parry, On the $ \beta $-expansions of real numbers, Acta Math. Acad. Sci. Hungar., 11 (1960), 401-416.  doi: 10.1007/BF02020954.  Google Scholar

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A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., 8 (1957), 477-493.  doi: 10.1007/BF02020331.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

J. C. Xiong, Hausdorff dimension of a chaotic set of shift of a symbolic space, Sci. China Ser. A, 38 (1995), 696-708.   Google Scholar

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

K. Dajani and C. Kraaikamp, Ergodic Theory of Numbers, Carus Mathematical Monographs, 29. Mathematical Association of America, Washington, DC, 2002.  Google Scholar

[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.  Google Scholar

[7]

K. Falconer, Fractal Geometry, , Mathematical foundations and applications. Third edition. John Wiley & Sons, Ltd., Chichester, 2014.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

F. Hofbauer, $\beta$-Shifts have unique maximal measure, Monatsh. Math., 85 (1978), 189-198.  doi: 10.1007/BF01534862.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

K.-S. Lau and L. Shu, The spectrum of Poincaré recurrence, Ergodic Theory Dynam. Systems, 28 (2008), 1917-1943.  doi: 10.1017/S0143385707001095.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

W. Parry, On the $ \beta $-expansions of real numbers, Acta Math. Acad. Sci. Hungar., 11 (1960), 401-416.  doi: 10.1007/BF02020954.  Google Scholar

[21]

A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., 8 (1957), 477-493.  doi: 10.1007/BF02020331.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

J. C. Xiong, Hausdorff dimension of a chaotic set of shift of a symbolic space, Sci. China Ser. A, 38 (1995), 696-708.   Google Scholar

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