Sensitivity, proximal extension and higher order almost automorphy

2016/05/04 by Xiangdong Ye, Tao Yu, Ye, Xiangdong +1 · 1 citation
Mathematics · #Mathematical Dynamics and Fractals #Advanced Topology and Set Theory #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.1605.01119

Abstract

Let (X,T) be a topological dynamical system, and F be a family of subsets of ℤ₊. (X,T) is strongly F-sensitive, if there is δ>0 such that for each non-empty open subset U, there are x,y∈ U with \n∈ℤ₊: d(Tⁿx,Tⁿy)>δ\\inF. Let F_t (resp. F_ip, F_fip) be consisting of thick sets (resp. IP-sets, subsets containing arbitrarily long finite IP-sets). The following Auslander-Yorke's type dichotomy theorems are obtained: (1) a minimal system is either strongly F_fip-sensitive or an almost one-to-one extension of its ∞-step nilfactor. (2) a minimal system is either strongly F_ip-sensitive or an almost one-to-one extension of its maximal distal factor. (3) a minimal system is either strongly F_t-sensitive or a proximal extension of its maximal distal factor.

Cited by

• Auslander-Yorke Dichotomy and Its Generalizations for Non-Autonomous Dynamical Systems

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