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doi: 10.3934/dcds.2021211
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On $ n $-tuplewise IP-sensitivity and thick sensitivity

Jian Li and  Yini Yang

Department of Mathematics, Shantou University, Shantou, Guangdong, 515063, China

Received  May 2021 Revised  November 2021 Early access January 2022

Fund Project: The authors were supported by NNSF of China (11771264, 11871188) and NSF of Guangdong Province (2018B030306024)

Let
$ (X,T) $
 be a topological dynamical system and
$ n\geq 2 $
. We say that
$ (X,T) $
 is
$ n $
-tuplewise IP-sensitive (resp.
$ n $
-tuplewise thickly sensitive) if there exists a constant
$ \delta>0 $
 with the property that for each non-empty open subset
$ U $
 of
$ X $
, there exist
$ x_1,x_2,\dotsc,x_n\in U $
 such that
$ \Bigl\{k\in \mathbb{N}\colon \min\limits_{1\le i<j\le n}d(T^k x_i,T^k x_j)>\delta\Bigr\} $
is an IP-set (resp. a thick set).
We obtain several sufficient and necessary conditions of a dynamical system to be
$ n $
-tuplewise IP-sensitive or
$ n $
-tuplewise thickly sensitive and show that any non-trivial weakly mixing system is
$ n $
-tuplewise IP-sensitive for all
$ n\geq 2 $
, while it is
$ n $
-tuplewise thickly sensitive if and only if it has at least
$ n $
 minimal points. We characterize two kinds of sensitivity by considering some kind of factor maps. We introduce the opposite side of pairwise IP-sensitivity and pairwise thick sensitivity, named (almost) pairwise IP
$ ^* $
-equicontinuity and (almost) pairwise syndetic equicontinuity, and obtain dichotomies results for them. In particular, we show that a minimal system is distal if and only if it is pairwise IP
$ ^* $
-equicontinuous. We show that every minimal system admits a maximal almost pairwise IP
$ ^* $
-equicontinuous factor and admits a maximal pairwise syndetic equicontinuous factor, and characterize them by the factor maps to their maximal distal factors.
Keywords: Sensitivity, equicontinuity, IP-set, thick set, minimal system, distal factor.
Mathematics Subject Classification: Primary: 37B05, 37B25.
Citation: Jian Li, Yini Yang. On $ n $-tuplewise IP-sensitivity and thick sensitivity. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021211
References:
[1]

E. Akin, Recurrence in Topological Dynamics, Furstenberg Families and Ellis Actions, The University Series in Mathematics, Plenum Press, New York, 1997. doi: 10.1007/978-1-4757-2668-8.  Google Scholar

[2]

J. Auslander, On the proximal relation in topological dynamics, Proc. Amer. Math. Soc., 11 (1960), 890-895.  doi: 10.1090/S0002-9939-1960-0164335-7.  Google Scholar

[3]

J. Auslander, Minimal Flows and Their Extensions, North-Holland Mathematics Studies, 153. Mathematical Notes, 122. North-Holland Publishing Co., Amsterdam, 1988.  Google Scholar

[4]

J. Auslander and J. A. Yorke, Interval maps, factors of maps, and chaos, Tohoku Math. J. (2), 32 (1980), 177-188.  doi: 10.2748/tmj/1178229634.  Google Scholar

[5]

L. Auslander, L. Green and F. Hahn, Flows on Homogeneous Spaces, Annals of Mathematics Studies, 53, Princeton University Press, Princeton, N.J. 1963.  Google Scholar

[6]

J. Clay, Variations on equicontinuity, Duke Math. J., 30 (1963), 423-431.  doi: 10.1215/S0012-7094-63-03045-X.  Google Scholar

[7]

J. P. Clay, Proximity relations in transformation groups, Trans. Amer. Math. Soc., 108 (1963), 88-96.  doi: 10.1090/S0002-9947-1963-0154269-3.  Google Scholar

[8]

R. Ellis and W. H. Gottschalk, Homomorphisms of transformation groups, Trans. Amer. Math. Soc., 94 (1960), 258-271.  doi: 10.1090/S0002-9947-1960-0123635-1.  Google Scholar

[9]

S. Fomin, On dynamical systems with a purely point spectrum (in Russian), Dokl. Akad. Nauk SSSR (N.S.), 77 (1951), 29-32.   Google Scholar

[10]

H. Furstenberg, Strict ergodicity and transformation of the torus, Amer. J. Math., 83 (1961), 573-601.  doi: 10.2307/2372899.  Google Scholar

[11]

H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Syst. Theory, 1 (1967), 1-49.  doi: 10.1007/BF01692494.  Google Scholar

[12]

H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, M. B. Porter Lectures, Princeton University Press, Princeton, N.J., 1981.  Google Scholar

[13]

W. H. He and Z. L. Zhou, A topologically mixing system whose measure center is a singleton, Acta Math. Sinica (Chin. Ser.), 45 (2002), 929-934.   Google Scholar

[14]

W. Huang, S. Kolyada and G. Zhang, Analogues of Auslander-Yorke theorems for multi-sensitivity, Ergodic Theory Dynam. Systems, 38 (2018), 651-665.  doi: 10.1017/etds.2016.48.  Google Scholar

[15]

W. Huang, D. Khilko, S. Kolyada, A. Peris and G. Zhang, Finite intersection property and dynamical compactness, J. Dynam. Differential Equations, 30 (2018), 1221-1245.  doi: 10.1007/s10884-017-9600-8.  Google Scholar

[16]

J. Li, Dynamical characterization of C-sets and its application, Fund. Math., 216 (2012), 259-286.  doi: 10.4064/fm216-3-4.  Google Scholar

[17]

J. Li, S. Tu and X. Ye, Mean equicontinuity and mean sensitivity, Ergodic Theory Dynam. Systems, 35 (2015), 2587-2612.  doi: 10.1017/etds.2014.41.  Google Scholar

[18]

J. Li and X. D. Ye, Recent development of chaos theory in topological dynamics, Acta Math. Sin. (Engl. Ser.), 32 (2016), 83-114.  doi: 10.1007/s10114-015-4574-0.  Google Scholar

[19]

J. Li and Y. Yang, Stronger versions of sensitivity for minimal group actions, Acta Math. Sin. (Engl. Ser.), 37 (2021), 1933-1946.  doi: 10.1007/s10114-021-0511-6.  Google Scholar

[20]

J. Li, X. Ye and T. Yu, Mean equicontinuity, complexity and applications, Discrete Contin. Dyn. Syst., 41 (2021), 359-393.  doi: 10.3934/dcds.2020167.  Google Scholar

[21]

T. K. S. Moothathu, Stronger forms of sensitivity for dynamical systems, Nonlinearity, 20 (2007), 2115-2126.  doi: 10.1088/0951-7715/20/9/006.  Google Scholar

[22]

F. Tan and R. Zhang, On $\mathcal{F}$-sensitive pairs, Acta Math. Sci. Ser. B (Engl. Ed.), 31 (2011), 1425-1435.  doi: 10.1016/S0252-9602(11)60328-7.  Google Scholar

[23]

J. de Vries, Elements of Topological Dynamics, Mathematics and its Applications, 257. Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-015-8171-4.  Google Scholar

[24]

J. Xiong, Chaos in a topologically transitive system, Sci. China Ser. A, 48 (2005), 929-939.  doi: 10.1007/BF02879075.  Google Scholar

[25]

X. Ye and T. Yu, Sensitivity, proximal extension and higher order almost automorphy, Trans. Amer. Math. Soc., 370 (2018), 3639-3662.  doi: 10.1090/tran/7100.  Google Scholar

show all references

References:
[1]

E. Akin, Recurrence in Topological Dynamics, Furstenberg Families and Ellis Actions, The University Series in Mathematics, Plenum Press, New York, 1997. doi: 10.1007/978-1-4757-2668-8.  Google Scholar

[2]

J. Auslander, On the proximal relation in topological dynamics, Proc. Amer. Math. Soc., 11 (1960), 890-895.  doi: 10.1090/S0002-9939-1960-0164335-7.  Google Scholar

[3]

J. Auslander, Minimal Flows and Their Extensions, North-Holland Mathematics Studies, 153. Mathematical Notes, 122. North-Holland Publishing Co., Amsterdam, 1988.  Google Scholar

[4]

J. Auslander and J. A. Yorke, Interval maps, factors of maps, and chaos, Tohoku Math. J. (2), 32 (1980), 177-188.  doi: 10.2748/tmj/1178229634.  Google Scholar

[5]

L. Auslander, L. Green and F. Hahn, Flows on Homogeneous Spaces, Annals of Mathematics Studies, 53, Princeton University Press, Princeton, N.J. 1963.  Google Scholar

[6]

J. Clay, Variations on equicontinuity, Duke Math. J., 30 (1963), 423-431.  doi: 10.1215/S0012-7094-63-03045-X.  Google Scholar

[7]

J. P. Clay, Proximity relations in transformation groups, Trans. Amer. Math. Soc., 108 (1963), 88-96.  doi: 10.1090/S0002-9947-1963-0154269-3.  Google Scholar

[8]

R. Ellis and W. H. Gottschalk, Homomorphisms of transformation groups, Trans. Amer. Math. Soc., 94 (1960), 258-271.  doi: 10.1090/S0002-9947-1960-0123635-1.  Google Scholar

[9]

S. Fomin, On dynamical systems with a purely point spectrum (in Russian), Dokl. Akad. Nauk SSSR (N.S.), 77 (1951), 29-32.   Google Scholar

[10]

H. Furstenberg, Strict ergodicity and transformation of the torus, Amer. J. Math., 83 (1961), 573-601.  doi: 10.2307/2372899.  Google Scholar

[11]

H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Syst. Theory, 1 (1967), 1-49.  doi: 10.1007/BF01692494.  Google Scholar

[12]

H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, M. B. Porter Lectures, Princeton University Press, Princeton, N.J., 1981.  Google Scholar

[13]

W. H. He and Z. L. Zhou, A topologically mixing system whose measure center is a singleton, Acta Math. Sinica (Chin. Ser.), 45 (2002), 929-934.   Google Scholar

[14]

W. Huang, S. Kolyada and G. Zhang, Analogues of Auslander-Yorke theorems for multi-sensitivity, Ergodic Theory Dynam. Systems, 38 (2018), 651-665.  doi: 10.1017/etds.2016.48.  Google Scholar

[15]

W. Huang, D. Khilko, S. Kolyada, A. Peris and G. Zhang, Finite intersection property and dynamical compactness, J. Dynam. Differential Equations, 30 (2018), 1221-1245.  doi: 10.1007/s10884-017-9600-8.  Google Scholar

[16]

J. Li, Dynamical characterization of C-sets and its application, Fund. Math., 216 (2012), 259-286.  doi: 10.4064/fm216-3-4.  Google Scholar

[17]

J. Li, S. Tu and X. Ye, Mean equicontinuity and mean sensitivity, Ergodic Theory Dynam. Systems, 35 (2015), 2587-2612.  doi: 10.1017/etds.2014.41.  Google Scholar

[18]

J. Li and X. D. Ye, Recent development of chaos theory in topological dynamics, Acta Math. Sin. (Engl. Ser.), 32 (2016), 83-114.  doi: 10.1007/s10114-015-4574-0.  Google Scholar

[19]

J. Li and Y. Yang, Stronger versions of sensitivity for minimal group actions, Acta Math. Sin. (Engl. Ser.), 37 (2021), 1933-1946.  doi: 10.1007/s10114-021-0511-6.  Google Scholar

[20]

J. Li, X. Ye and T. Yu, Mean equicontinuity, complexity and applications, Discrete Contin. Dyn. Syst., 41 (2021), 359-393.  doi: 10.3934/dcds.2020167.  Google Scholar

[21]

T. K. S. Moothathu, Stronger forms of sensitivity for dynamical systems, Nonlinearity, 20 (2007), 2115-2126.  doi: 10.1088/0951-7715/20/9/006.  Google Scholar

[22]

F. Tan and R. Zhang, On $\mathcal{F}$-sensitive pairs, Acta Math. Sci. Ser. B (Engl. Ed.), 31 (2011), 1425-1435.  doi: 10.1016/S0252-9602(11)60328-7.  Google Scholar

[23]

J. de Vries, Elements of Topological Dynamics, Mathematics and its Applications, 257. Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-015-8171-4.  Google Scholar

[24]

J. Xiong, Chaos in a topologically transitive system, Sci. China Ser. A, 48 (2005), 929-939.  doi: 10.1007/BF02879075.  Google Scholar

[25]

X. Ye and T. Yu, Sensitivity, proximal extension and higher order almost automorphy, Trans. Amer. Math. Soc., 370 (2018), 3639-3662.  doi: 10.1090/tran/7100.  Google Scholar

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