Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for <inline-formula><tex-math id="M1">\begin{document} \beta \end{document}</tex-math></inline-formula>-transformation

Previous Article
Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains
DCDS Home
This Issue
Next Article
The unique measure of maximal entropy for a compact rank one locally CAT(0) space

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

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 and Continuous Dynamical Systems, 2021, 41 (2) : 525-536. doi: 10.3934/dcds.2020267

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.

[1]	Jakub Šotola. Relationship between Li-Yorke chaos and positive topological sequence entropy in nonautonomous dynamical systems. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5119-5128. doi: 10.3934/dcds.2018225
[2]	Ghassen Askri. Li-Yorke chaos for dendrite maps with zero topological entropy and ω-limit sets. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 2957-2976. doi: 10.3934/dcds.2017127
[3]	Daniel Gonçalves, Bruno Brogni Uggioni. Li-Yorke Chaos for ultragraph shift spaces. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2347-2365. doi: 10.3934/dcds.2020117
[4]	JinHyon Kim, HyonHui Ju, WiJong An. Inheritance of $ {\mathscr F}- $chaos and $ {\mathscr F}- $sensitivities under an iteration for non-autonomous discrete systems. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022053
[5]	Dominik Kwietniak. Topological entropy and distributional chaos in hereditary shifts with applications to spacing shifts and beta shifts. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2451-2467. doi: 10.3934/dcds.2013.33.2451
[6]	María Anguiano, Alain Haraux. The $\varepsilon$-entropy of some infinite dimensional compact ellipsoids and fractal dimension of attractors. Evolution Equations and Control Theory, 2017, 6 (3) : 345-356. doi: 10.3934/eect.2017018
[7]	Pak Tung Ho. Prescribing the $ Q' $-curvature in three dimension. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 2285-2294. doi: 10.3934/dcds.2019096
[8]	Thomas French. Follower, predecessor, and extender set sequences of $ \beta $-shifts. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4331-4344. doi: 10.3934/dcds.2019175
[9]	Mohd Raiz, Amit Kumar, Vishnu Narayan Mishra, Nadeem Rao. Dunkl analogue of Sz$ \acute{a} $sz-Schurer-Beta operators and their approximation behaviour. Mathematical Foundations of Computing, 2022 doi: 10.3934/mfc.2022007
[10]	Victor Vargas. On involution kernels and large deviations principles on $ \beta $-shifts. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2699-2718. doi: 10.3934/dcds.2021208
[11]	Magdalena Foryś-Krawiec, Jiří Kupka, Piotr Oprocha, Xueting Tian. On entropy of $ \Phi $-irregular and $ \Phi $-level sets in maps with the shadowing property. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1271-1296. doi: 10.3934/dcds.2020317
[12]	Liang Ding, Weimin Han. Morozov's discrepancy principle for $ \alpha\ell_1-\beta\ell_2 $ sparsity regularization. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022035
[13]	Eric A. Carlen, Maria C. Carvalho, Jonathan Le Roux, Michael Loss, Cédric Villani. Entropy and chaos in the Kac model. Kinetic and Related Models, 2010, 3 (1) : 85-122. doi: 10.3934/krm.2010.3.85
[14]	Adam Kanigowski, Federico Rodriguez Hertz, Kurt Vinhage. On the non-equivalence of the Bernoulli and $ K$ properties in dimension four. Journal of Modern Dynamics, 2018, 13: 221-250. doi: 10.3934/jmd.2018019
[15]	Yu-Zhao Wang. $ \mathcal{W}$-Entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2441-2454. doi: 10.3934/cpaa.2018116
[16]	Yanheng Ding, Xiaojing Dong, Qi Guo. On multiplicity of semi-classical solutions to nonlinear Dirac equations of space-dimension $ n $. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4105-4123. doi: 10.3934/dcds.2021030
[17]	Zijun Chen, Shengkun Wu. Local well-posedness for the Zakharov system in dimension $ d = 2, 3 $. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4307-4319. doi: 10.3934/cpaa.2021161
[18]	Anhui Gu. Weak pullback mean random attractors for non-autonomous $ p $-Laplacian equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3863-3878. doi: 10.3934/dcdsb.2020266
[19]	Yixiao Qiao, Xiaoyao Zhou. Zero sequence entropy and entropy dimension. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 435-448. doi: 10.3934/dcds.2017018
[20]	Pak Tung Ho. Prescribing $ Q $-curvature on $ S^n $ in the presence of symmetry. Communications on Pure and Applied Analysis, 2020, 19 (2) : 715-722. doi: 10.3934/cpaa.2020033

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Yuanfen Xiao
on Google Scholar
Yuanfen Xiao

2020 Impact Factor: 1.392

Tools

Article outline

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Yuanfen Xiao
on Google Scholar
Yuanfen Xiao

[Back to Top]