Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Sensitivity, proximal extension and higher order almost automorphy


Authors: Xiangdong Ye and Tao Yu
Journal: Trans. Amer. Math. Soc. 370 (2018), 3639-3662
MSC (2010): Primary 37B05; Secondary 54H20
DOI: https://doi.org/10.1090/tran/7100
Published electronically: November 15, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ (X,T)$ be a topological dynamical system, and $ \mathcal {F}$ be a family of subsets of $ \mathbb{Z}_+$. $ (X,T)$ is strongly $ \mathcal {F}$-sensitive if there is $ \delta >0$ such that for each non-empty open subset $ U$ there are $ x,y\in U$ with $ \{n\in \mathbb{Z}_+: d(T^nx,T^ny)>\delta \}\in \mathcal {F}$. Let $ \mathcal {F}_t$ (resp. $ \mathcal {F}_{ip}$, $ \mathcal {F}_{fip}$) consist of thick sets (resp. IP-sets, subsets containing arbitrarily long finite IP-sets).

The following Auslander-Yorke's type dichotomy theorems are obtained: (1) a minimal system is either strongly $ \mathcal {F}_{fip}$-sensitive or an almost one-to-one extension of its $ \infty $-step nilfactor; (2) a minimal system is either strongly $ \mathcal {F}_{ip}$-sensitive or an almost one-to-one extension of its maximal distal factor; (3) a minimal system is either strongly $ \mathcal {F}_{t}$-sensitive or a proximal extension of its maximal distal factor.


References [Enhancements On Off] (What's this?)

  • [1] Ethan Akin and Sergiĭ Kolyada, Li-Yorke sensitivity, Nonlinearity 16 (2003), no. 4, 1421-1433. MR 1986303, https://doi.org/10.1088/0951-7715/16/4/313
  • [2] Joseph Auslander, On the proximal relation in topological dynamics, Proc. Amer. Math. Soc. 11 (1960), 890-895. MR 0164335, https://doi.org/10.2307/2034433
  • [3] 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
  • [4] J. Auslander and Brindell Horelick, Regular minimal sets. II. The proximally equicontinuous case, Compositio Math. 22 (1970), 203-214. MR 0267554
  • [5] Joseph Auslander and James A. Yorke, Interval maps, factors of maps, and chaos, Tôhoku Math. J. (2) 32 (1980), no. 2, 177-188. MR 580273, https://doi.org/10.2748/tmj/1178229634
  • [6] Vitaly Bergelson, Ultrafilters, IP sets, dynamics, and combinatorial number theory, Ultrafilters across mathematics, Contemp. Math., vol. 530, Amer. Math. Soc., Providence, RI, 2010, pp. 23-47. MR 2757532, https://doi.org/10.1090/conm/530/10439
  • [7] 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, DOI 10.1017/etds.2015.83. MR 3645516.
  • [8] Pandeng Dong, Sebastián Donoso, Alejandro Maass, Song Shao, and Xiangdong Ye, Infinite-step nilsystems, independence and complexity, Ergodic Theory Dynam. Systems 33 (2013), no. 1, 118-143. MR 3009105, https://doi.org/10.1017/S0143385711000861
  • [9] Pandeng Dong, Song Shao, and Xiangdong Ye, Product recurrent properties, disjointness and weak disjointness, Israel J. Math. 188 (2012), 463-507. MR 2897741, https://doi.org/10.1007/s11856-011-0128-z
  • [10] Tomasz Downarowicz and Eli Glasner, Isomorphic extensions and applications, Topol. Methods Nonlinear Anal. 48 (2016), no. 1, 321-338. MR 3586277, https://doi.org/10.12775/TMNA.2016.050
  • [11] Robert Ellis, Shmuel Glasner, and Leonard Shapiro, Proximal-isometric ( $ \mathcal{P}\mathcal{J} $) flows, Advances in Math. 17 (1975), no. 3, 213-260. MR 0380755, https://doi.org/10.1016/0001-8708(75)90093-6
  • [12] H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, N.J., 1981. M. B. Porter Lectures. MR 603625
  • [13] H. Furstenberg and Y. Katznelson, An ergodic Szemerédi theorem for IP-systems and combinatorial theory, J. Analyse Math. 45 (1985), 117-168. MR 833409, https://doi.org/10.1007/BF02792547
  • [14] J. Gillis, Note on a Property of Measurable Sets, J. London Math. Soc. S1-11, no. 2, 139. MR 1574762, https://doi.org/10.1112/jlms/s1-11.2.139
  • [15] Eli Glasner, Book Review: Minimal flows and their extensions, Bull. Amer. Math. Soc. (N.S.) 21 (1989), no. 2, 316-319. MR 1567809, https://doi.org/10.1090/S0273-0979-1989-15843-6
  • [16] Eli Glasner, Yonatan Gutman, and Xiangdong Ye, Higher order regionally proximal equivalence relations for general group actions, arXiv:1706.07227[math.DS].
  • [17] Eli Glasner and Benjamin Weiss, Sensitive dependence on initial conditions, Nonlinearity 6 (1993), no. 6, 1067-1075. MR 1251259
  • [18] S. Glasner and B. Weiss, On the construction of minimal skew products, Israel J. Math. 34 (1979), no. 4, 321-336 (1980). MR 570889, https://doi.org/10.1007/BF02760611
  • [19] Bernard Host, Bryna Kra, and Alejandro Maass, Nilsequences and a structure theorem for topological dynamical systems, Adv. Math. 224 (2010), no. 1, 103-129. MR 2600993, https://doi.org/10.1016/j.aim.2009.11.009
  • [20] Wen Huang, Danylo Khilko, Sergiĭ Kolyada, and Guohua Zhang, Dynamical compactness and sensitivity, J. Differential Equations 260 (2016), no. 9, 6800-6827. MR 3461085, https://doi.org/10.1016/j.jde.2016.01.011
  • [21] Wen Huang, Sergiĭ Kolyada, and Guohua Zhang, Analogues of Auslander-Yorke theorems for multi-sensitivity, arXiv:1509.08818[math.DS], Ergodic Theory Dynam. Systems, to appear.
  • [22] Wen Huang, Ping Lu, and Xiangdong Ye, Measure-theoretical sensitivity and equicontinuity, Israel J. Math. 183 (2011), 233-283. MR 2811160, https://doi.org/10.1007/s11856-011-0049-x
  • [23] Wen Huang, Song Shao, and Xiangdong Ye, Nil Bohr-sets and almost automorphy of higher order, Mem. Amer. Math. Soc. 241 (2016), no. 1143, v+83. MR 3476203, https://doi.org/10.1090/memo/1143
  • [24] Jian Li, Dynamical characterization of C-sets and its application, Fund. Math. 216 (2012), no. 3, 259-286. MR 2890544, https://doi.org/10.4064/fm216-3-4
  • [25] 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, https://doi.org/10.1155/2014/769523
  • [26] Jian Li, Siming Tu, and Xiangdong Ye, Mean equicontinuity and mean sensitivity, Ergodic Theory Dynam. Systems 35 (2015), no. 8, 2587-2612. MR 3456608
  • [27] Jian Li and Xiang Dong Ye, Recent development of chaos theory in topological dynamics, Acta Math. Sin. (Engl. Ser.) 32 (2016), no. 1, 83-114. MR 3431162, https://doi.org/10.1007/s10114-015-4574-0
  • [28] 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, https://doi.org/10.1155/2014/583431
  • [29] Leonard Shapiro, Proximality in minimal transformation groups, Proc. Amer. Math. Soc. 26 (1970), 521-525. MR 0266183, https://doi.org/10.2307/2037372
  • [30] T. K. Subrahmonian Moothathu, Stronger forms of sensitivity for dynamical systems, Nonlinearity 20 (2007), no. 9, 2115-2126. MR 2351026, https://doi.org/10.1088/0951-7715/20/9/006
  • [31] 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
  • [32] Song Shao and Xiangdong Ye, Regionally proximal relation of order $ d$ is an equivalence one for minimal systems and a combinatorial consequence, Adv. Math. 231 (2012), no. 3-4, 1786-1817. MR 2964624, https://doi.org/10.1016/j.aim.2012.07.012

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 37B05, 54H20

Retrieve articles in all journals with MSC (2010): 37B05, 54H20


Additional Information

Xiangdong Ye
Affiliation: Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences, Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, People’s Republic of China
Email: yexd@ustc.edu.cn

Tao Yu
Affiliation: Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences, Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, People’s Republic of China
Email: ytnuo@mail.ustc.edu.cn

DOI: https://doi.org/10.1090/tran/7100
Keywords: Sensitivity, minimality, infinite step nilfactor, distal factor
Received by editor(s): May 7, 2016
Received by editor(s) in revised form: August 19, 2016
Published electronically: November 15, 2017
Additional Notes: The authors were supported by NNSF of China (11371339, 11431012, 11571335).
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society