Open Access
Original Articles
Measure-Theoretical Sensitivity and Scrambled Sets Via Furstenberg Families
- Published Online:
- 03 Jun 2013
- Article type:
- Original Articles
- Language:
- EN
- Article no.:
- 1726037X.2009.10698558
- Pages:
- 1–12
Abstract
For an invariant measure μ in a topological dynamic system, notions of F-μ-pairwise sensitivity and F-μ-sensitivity are introduced and investigated, where F is a Furstenberg family. It is shown that F-μ-pairwise sensitivity is equivalent to F-μ-sensitivity, when F is a filterdual and is extensively compatible to the 2-fold product system. Moreover, it turns out that if (X, B, μ, f) is a measure-preserving system of weakly mixing and the support of μ is X, then there exists a positive c such that (X, f) has an (M* (c), M* (0))- scrambled set with full induced outer measure by μ. M* (c) denotes the family of all subsets of ℤ+ whose lower density is not less than c, and M* (0) denotes the family of all subsets of ℤ+ whose lower density is non-zero.
Keywords
References
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