Measure-Theoretical Sensitivity and Scrambled Sets via Furstenberg Families

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.