Furstenberg Families and Sensitivity
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(f...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
2010-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2010/649348 |
| Summary: | 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. |
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| ISSN: | 1026-0226 1607-887X |