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

Abstract

In this paper, $d$-step almost automorphic systems are studied for $d\in \mathbb{N}$, which are the generalization of the classical almost automorphic ones. For a minimal topological dynamical system $(X,T)$ it is shown that the condition $x\in X$ is $d$-step almost automorphic can be characterized via various subsets of $\mathbb{Z}$ including the dual sets of $d$-step Poincaré and Birkhoff recurrence sets, and Nil${}_d$ Bohr${}_0$-sets by considering $N(x,V)=\{n\in \mathbb{Z}:T^{n}x\in V\}$, where $V$ is an arbitrary neighborhood of $x$. Moreover, it turns out that the condition $(x,y)\in X\times X$ is regionally proximal of order $d$ can also be characterized via various subsets of $\mathbb{Z}$ including $d$-step Poincaré and Birkhoff recurrence sets, $S{G}_d$ sets, the dual sets of Nil${}_d$ Bohr${}_0$-sets, and others by considering $N(x,U)=\{n\in \mathbb{Z}:T^{n}x\in U\}$, where $U$ is an arbitrary neighborhood of $y$.

37 pages

References in corpus (1)

• Nil Bohr${}_0$-sets, Poincaré recurrence and generalized polynomials

Cited by in corpus (1)

• Dynamical cubes and a criteria for systems having products extensions