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

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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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