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Journal of Dynamical Systems and Geometric Theories

(Since 2002)
Open Access ·Peer-reviewed·ISSN (Online): 2169-0057·ISSN (Print): 1726-037X

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Half-Yearly Journal: Publishes research on dynamical systems and geometry, including Random Dynamical Systems, hyperbolic behaviour, complex geometry and Ergodic theory.

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Journal of Dynamical Systems and Geometric Theories
Open Access Original Articles

Measure-Theoretical Sensitivity and Scrambled Sets via Furstenberg Families

pp. 185–198Vol. 6Issue 2June 2013DOI: 10.1080/1726037X.2008.10698556XML
Published Online:
03 Jun 2013
Article type:
Original Articles
Language:
EN
Article no.:
1726037X.2008.10698556
Pages:
185–198

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

SensitivityScrambled setFurstenberg familyWeakly mixing

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