Higher order almost automorphy, recurrence sets and the regionally proximal relation

2011/10/30 by Huang, Wen, Shao, Song, Ye, Xiangdong
#37B05 #37B20 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1110.6599

Abstract

In this paper, d-step almost automorphic systems are studied for d∈\N, which are the generalization of the classical almost automorphic ones. For a minimal topological dynamical system (X,T) it is shown that the condition x∈ X is d-step almost automorphic can be characterized via various subsets of \Z including the dual sets of d-step Poincaré and Birkhoff recurrence sets, and Nil_d Bohr₀-sets by considering N(x,V)=\n∈\Z: Tⁿx∈ V\, where V is an arbitrary neighborhood of x. Moreover, it turns out that the condition (x,y)∈ X× X is regionally proximal of order d can also be characterized via various subsets of \Z including d-step Poincaré and Birkhoff recurrence sets, SG_d sets, the dual sets of Nil_d Bohr₀-sets, and others by considering N(x,U)=\n∈\Z: Tⁿx∈ U\, where U is an arbitrary neighborhood of y.

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