Furstenberg Families and Sensitivity

• Main Authors: Huoyun Wang, Jincheng Xiong, Feng Tan
• Format: Article
• Language: English
• Published: Hindawi Limited 2010-01-01
• Series: Discrete Dynamics in Nature and Society
• Online Access: http://dx.doi.org/10.1155/2010/649348

We introduce and study some concepts of sensitivity via Furstenberg families. A dynamical system (X,f) is ℱ-sensitive if there exists a positive ε such that for every x∈X and every open neighborhood U of x there exists y∈U such that the pair (x,y) is not ℱ-ε-asymptotic; that is, the time set {n:d(fn(x),fn(y))>ε} belongs to ℱ, where ℱ is a Furstenberg family. A dynamical system (X,f) is (ℱ1, ℱ2)-sensitive if there is a positive ε such that every x∈X is a limit of points y∈X such that the pair (x,y) is ℱ1-proximal but not ℱ2-ε-asymptotic; that is, the time set {n:d(fn(x),fn(y))<δ} belongs to ℱ1 for any positive δ but the time set {n:d(fn(x),fn(y))>ε} belongs to ℱ2, where ℱ1 and ℱ2 are Furstenberg families.