ON F-SENSITIVE PAIRS

doi:10.1016/S0252-9602(11)60328-7

数学物理学报(英文版) ›› 2011, Vol. 31 ›› Issue (4): 1425-1435.doi: 10.1016/S0252-9602(11)60328-7

ON F-SENSITIVE PAIRS

谭枫|张瑞丰

School of mathematical science, South China Normal University, Guangzhou 510631, China； School of mathematics, Hefei University of Technology, Hefei 230009, China

收稿日期:2009-07-30 修回日期:2010-03-15 出版日期:2011-07-20 发布日期:2011-07-20
基金资助:
The first author is supported by NSFC (10771079;
10871186; 11071084; 11026095) and NSF of Guangdong Province (10451063101006332). The second author is
partially supported by NSFC (11001071) and Hefei University of Technology (GDBJ2008-024; 2010HGXJ0200).

ON F-SENSITIVE PAIRS

TAN Feng, ZHANG Rui-Feng

School of mathematical science, South China Normal University, Guangzhou 510631, China； School of mathematics, Hefei University of Technology, Hefei 230009, China

Received:2009-07-30 Revised:2010-03-15 Online:2011-07-20 Published:2011-07-20
Supported by:
The first author is supported by NSFC (10771079;
10871186; 11071084; 11026095) and NSF of Guangdong Province (10451063101006332). The second author is
partially supported by NSFC (11001071) and Hefei University of Technology (GDBJ2008-024; 2010HGXJ0200).

摘要/Abstract

摘要：

In the present paper, we define sensitive pairs via Furstenberg families and
discuss the relation of three definitions: sensitivity, F-sensitivity and F-sensitive pairs,
see Theorem 1. For transitive systems, we give some su?cient conditions to ensure the
existence of F-sensitive pairs. In particular, each non-minimal E system (M system, P sys-
tem) has positive lower density (Fs, Fr resp.)-sensitive pairs almost everywhere. Moreover,
each non-minimal M system is Fts-sensitive. Finally, by some examples we show that: (1)
F-sensitivity can not imply the existence of F-sensitive pairs. That means there exists an
F-sensitive system, which has no F-sensitive pairs. (2) There is no immediate relation be-
tween the existence of sensitive pairs and Li-Yorke chaos, i.e., there exists a system (X,f)
without Li-Yorke scrambled pairs, which has κB-sensitive pairs almost everywhere. (3)
If the system (G,f) is sensitive, where G is a finite graph, then it has κB-sensitive pairs
almost everywhere.

关键词: sensitive pair, Furstenberg family, transitive system

Abstract:

Key words: sensitive pair, Furstenberg family, transitive system

中图分类号:

54H20

谭枫, 张瑞丰. ON F-SENSITIVE PAIRS[J]. 数学物理学报(英文版), 2011, 31(4): 1425-1435.

TAN Feng, ZHANG Rui-Feng. ON F-SENSITIVE PAIRS[J]. Acta mathematica scientia,Series B, 2011, 31(4): 1425-1435.

参考文献

[1] Glasner E, Weiss B. Sensitive dependence on initial conditions. Nonlinearity, 1993, 6: 1067–1075

[2] Xiong J. Chaos in topological transitive systems. Science in China Series A, 2005, 48: 929–939

[3] Shao S, Ye X, Zhang R. Sensitivity and regionally proximal relation in minimal systems. Science in China
Series A, 2008, 51: 987–994

[4] Ye X, Zhang R. On sensitive sets in topological dynamics. Nonlinearity, 2008, 21: 1601–1620

[5] Akin E, Auslander J, Berg K. When is a transitive map chaotic//Convergence in Ergodic Theory and
Probability (Columbus, OH, 1993). Ohio State Univ Math Res Inst Publ, 5. Berlin: de Gruyter, 1996:
25–40

[6] Huang W, Ye X. An Explicit scattering, non-weakly mixing example and weak disjointness. Nonlinearity,
2002, 15: 825–846

[7] Moothathu T K S. Stronger forms of sensitivity for dynamical systems. Nonlinearity, 2007, 20: 2115–2126

[8] Akin E. Recurrence in Topological Dynamics: Furstenberg and Ellis actions. New York: Plenum Press,
1997

[9] Birkho? G. Proof of the ergodic theorem. Proc Nat Acad Sci USA, 1931, 17: 656–660

[10] Ye X, Huang W, Shao S. An Introduction to Topological Dynamical Systems. Beijing: Science Press, 2008

[11] Akin E, Glasner E. Residual properties and almost equicontinuity. J D’Analyse Math, 2001, 84: 243–286

[12] Blanchard F, Huang W, Snoha L. Topological size of scrambled sets. Colloq Math, 2008, 110: 293–361

[13] Xiong J, L¨u J, Tan F. Furstenberg family and chaos. Science in China Series A, 2007, 50: 1325–1333

[14] Mycielski J. Independent sets in topological algebras. Fund Math, 1964, 55: 139–147

[15] Shao S. Proximity and distality via Furstenberg families. Topol Appl, 2006, 153: 2055–2072

[16] He W, Zhou Z. A topologically mixing system with its measure center being a singleton. Acta Math Sinica,
2002, 45(5): 929–934

[17] Blanchard F, Glasner E, Kolyada S, Maass A. On Li-Yorke pairs. J Reine Angew Math, 2002, 547: 51–68

[18] Kato H. Chaos of continuum-wise expansive homeomorphisms and dynamical properties of sensitive maps
of graphs. Paci J Math, 1996, 175(1): 93–116

ON F-SENSITIVE PAIRS

ON F-SENSITIVE PAIRS

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 1

Metrics

本文评价

推荐阅读 10