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June  2022, 42(6): 2775-2793. doi: 10.3934/dcds.2021211

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 Published  June 2022 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 and Continuous Dynamical Systems, 2022, 42 (6) : 2775-2793. 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.

[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.

[3]

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

[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.

[5]

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

[6]

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

[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.

[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.

[9]

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

[10]

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

[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.

[12]

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

[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. 

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

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.

[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.

[3]

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

[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.

[5]

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

[6]

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

[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.

[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.

[9]

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

[10]

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

[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.

[12]

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

[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. 

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

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