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

• 论文 • 上一篇    下一篇

ON F-SENSITIVE PAIRS

谭枫|张瑞丰   

  1. 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   

  1. 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:

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:

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.

Key words: sensitive pair, Furstenberg family, transitive system

中图分类号: 

  • 54H20