Discrete Dynamics in Nature and Society
Volume 2010 (2010), Article ID 649348, 12 pages
Research Article

Furstenberg Families and Sensitivity

1Department of Mathematics, Guangzhou University, Guangzhou 510006, China
2Department of Mathematics, South China Normal University, Guangzhou 526061, China

Received 31 August 2009; Revised 17 November 2009; Accepted 22 January 2010

Academic Editor: Yong Zhou

Copyright © 2010 Huoyun Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We introduce and study some concepts of sensitivity via Furstenberg families. A dynamical system (X,f) is -sensitive if there exists a positive ε such that for every xX and every open neighborhood U of x there exists yU such that the pair (x,y) is not -ε-asymptotic; that is, the time set {n:d(fn(x),fn(y))>ε} belongs to , where is a Furstenberg family. A dynamical system (X,f) is (1, 2)-sensitive if there is a positive ε such that every xX is a limit of points yX such that the pair (x,y) is 1-proximal but not 2-ε-asymptotic; that is, the time set {n:d(fn(x),fn(y))<δ} belongs to 1 for any positive δ but the time set {n:d(fn(x),fn(y))>ε} belongs to 2, where 1 and 2 are Furstenberg families.