Auslander-Yorke type dichotomy theorems for stronger versions of 𝑟-sensitivity

Liu, Kairan; Zhou, Xiaomin

doi:10.1090/proc/14435

Auslander-Yorke type dichotomy theorems for stronger versions of $r$-sensitivity
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by Kairan Liu and Xiaomin Zhou

Proc. Amer. Math. Soc. 147 (2019), 2609-2617

DOI: https://doi.org/10.1090/proc/14435

Published electronically: February 20, 2019

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Abstract:

In this paper, for $r\in \mathbb {N}$ with $r\geq 2$ we consider several stronger versions of $r$-sensitivity and measure-theoretical $r$-sensitivities by analyzing subsets of non-negative integers, for which the $r$-sensitivity occurs. We obtain an Auslander-Yorke type dichotomy theorem: a minimal topological dynamical system is either thickly $r$-sensitive or an almost $m$-to-one extension of its maximal equicontinuous factor for some $m\in \{1,\cdots , r-1\}$.

References

Christophe Abraham, Gérard Biau, and Benoît Cadre, On Lyapunov exponent and sensitivity, J. Math. Anal. Appl. 290 (2004), no. 2, 395–404. MR 2033030, DOI 10.1016/j.jmaa.2003.10.029
Domenico Aiello, Hansheng Diao, Zhou Fan, Daniel O. King, Jessica Lin, and Cesar E. Silva, Measurable time-restricted sensitivity, Nonlinearity 25 (2012), no. 12, 3313–3325. MR 2993055, DOI 10.1088/0951-7715/25/12/3313
Joseph Auslander, Minimal flows and their extensions, North-Holland Mathematics Studies, vol. 153, North-Holland Publishing Co., Amsterdam, 1988. Notas de Matemática [Mathematical Notes], 122. MR 956049
Joseph Auslander and James A. Yorke, Interval maps, factors of maps, and chaos, Tohoku Math. J. (2) 32 (1980), no. 2, 177–188. MR 580273, DOI 10.2748/tmj/1178229634
F. Blanchard, B. Host, and A. Maass, Topological complexity, Ergodic Theory Dynam. Systems 20 (2000), no. 3, 641–662. MR 1764920, DOI 10.1017/S0143385700000341
Benoît Cadre and Pierre Jacob, On pairwise sensitivity, J. Math. Anal. Appl. 309 (2005), no. 1, 375–382. MR 2154050, DOI 10.1016/j.jmaa.2005.01.061
Robert Ellis and W. H. Gottschalk, Homomorphisms of transformation groups, Trans. Amer. Math. Soc. 94 (1960), 258–271. MR 123635, DOI 10.1090/S0002-9947-1960-0123635-1
Felipe García-Ramos, Weak forms of topological and measure-theoretical equicontinuity: relationships with discrete spectrum and sequence entropy, Ergodic Theory Dynam. Systems 37 (2017), no. 4, 1211–1237. MR 3645516, DOI 10.1017/etds.2015.83
Eli Glasner and Benjamin Weiss, Sensitive dependence on initial conditions, Nonlinearity 6 (1993), no. 6, 1067–1075. MR 1251259
John Guckenheimer, Sensitive dependence to initial conditions for one-dimensional maps, Comm. Math. Phys. 70 (1979), no. 2, 133–160. MR 553966
Jared Hallett, Lucas Manuelli, and Cesar E. Silva, On Li-Yorke measurable sensitivity, Proc. Amer. Math. Soc. 143 (2015), no. 6, 2411–2426. MR 3326024, DOI 10.1090/S0002-9939-2015-12430-6
Wen Huang, Danylo Khilko, Sergiĭ Kolyada, and Guohua Zhang, Dynamical compactness and sensitivity, J. Differential Equations 260 (2016), no. 9, 6800–6827. MR 3461085, DOI 10.1016/j.jde.2016.01.011
Wen Huang, Sergiĭ Kolyada, and Guohua Zhang, Analogues of Auslander-Yorke theorems for multi-sensitivity, Ergodic Theory Dynam. Systems 38 (2018), no. 2, 651–665. MR 3774836, DOI 10.1017/etds.2016.48
Wen Huang, Ping Lu, and Xiangdong Ye, Measure-theoretical sensitivity and equicontinuity, Israel J. Math. 183 (2011), 233–283. MR 2811160, DOI 10.1007/s11856-011-0049-x
Wen Huang and Xiangdong Ye, Devaney’s chaos or 2-scattering implies Li-Yorke’s chaos, Topology Appl. 117 (2002), no. 3, 259–272. MR 1874089, DOI 10.1016/S0166-8641(01)00025-6
Jennifer James, Thomas Koberda, Kathryn Lindsey, Cesar E. Silva, and Peter Speh, Measurable sensitivity, Proc. Amer. Math. Soc. 136 (2008), no. 10, 3549–3559. MR 2415039, DOI 10.1090/S0002-9939-08-09294-0
Jian Li, Measure-theoretic sensitivity via finite partitions, Nonlinearity 29 (2016), no. 7, 2133–2144. MR 3521642, DOI 10.1088/0951-7715/29/7/2133
Risong Li and Yuming Shi, Stronger forms of sensitivity for measure-preserving maps and semiflows on probability spaces, Abstr. Appl. Anal. , posted on (2014), Art. ID 769523, 10. MR 3208565, DOI 10.1155/2014/769523
Heng Liu, Li Liao, and Lidong Wang, Thickly syndetical sensitivity of topological dynamical system, Discrete Dyn. Nat. Soc. , posted on (2014), Art. ID 583431, 4. MR 3200824, DOI 10.1155/2014/583431
T. K. Subrahmonian Moothathu, Stronger forms of sensitivity for dynamical systems, Nonlinearity 20 (2007), no. 9, 2115–2126. MR 2351026, DOI 10.1088/0951-7715/20/9/006
David Ruelle, Dynamical systems with turbulent behavior, Mathematical problems in theoretical physics (Proc. Internat. Conf., Univ. Rome, Rome, 1977) Lecture Notes in Phys., vol. 80, Springer, Berlin-New York, 1978, pp. 341–360. MR 518445
Song Shao, XiangDong Ye, and RuiFeng Zhang, Sensitivity and regionally proximal relation in minimal systems, Sci. China Ser. A 51 (2008), no. 6, 987–994. MR 2410978, DOI 10.1007/s11425-008-0012-4
Feng Tan and Ruifeng Zhang, On $\scr F$-sensitive pairs, Acta Math. Sci. Ser. B (Engl. Ed.) 31 (2011), no. 4, 1425–1435. MR 2856453, DOI 10.1016/S0252-9602(11)60328-7
William A. Veech, The equicontinuous structure relation for minimal Abelian transformation groups, Amer. J. Math. 90 (1968), 723–732. MR 232377, DOI 10.2307/2373480
Huoyun Wang, Jincheng Xiong, and Feng Tan, Furstenberg families and sensitivity, Discrete Dyn. Nat. Soc. , posted on (2010), Art. ID 649348, 12. MR 2600324, DOI 10.1155/2010/649348
Hongying Wu and Huoyun Wang, Measure-theoretical sensitivity and scrambled sets via Furstenberg families, J. Dyn. Syst. Geom. Theor. 7 (2009), no. 1, 1–12. MR 2555894, DOI 10.1080/1726037X.2009.10698558
Jincheng Xiong, Chaos in a topologically transitive system, Sci. China Ser. A 48 (2005), no. 7, 929–939. MR 2179632, DOI 10.1360/04ys0120
Xiangdong Ye and Tao Yu, Sensitivity, proximal extension and higher order almost automorphy, Trans. Amer. Math. Soc. 370 (2018), no. 5, 3639–3662. MR 3766861, DOI 10.1090/tran/7100
Xiangdong Ye and Ruifeng Zhang, On sensitive sets in topological dynamics, Nonlinearity 21 (2008), no. 7, 1601–1620. MR 2425336, DOI 10.1088/0951-7715/21/7/012
Tao Yu, Measurable sensitivity via Furstenberg families, Discrete Contin. Dyn. Syst. 37 (2017), no. 8, 4543–4563. MR 3642276, DOI 10.3934/dcds.2017194
Yong Zou, Stronger version sensitivity, almost finite to one extension and maximal pattern entropy, Commun. Math. Stat. 5 (2017), no. 2, 123–139. MR 3665309, DOI 10.1007/s40304-017-0104-y