Abstract
Let T: X → X be a continuous map of a compact metric space X. A point x ∈ X is called Banach recurrent point if for all neighborhood V of x, {n ∈ ℕ: Tn(x) ∈ V} has positive upper Banach density. Denote by Tr(T), W(T), QW(T) and BR(T) the sets of transitive points, weakly almost periodic points, quasi-weakly almost periodic points and Banach recurrent points of (X, T). If (X, T) has the specification property, then we show that every transitive point is Banach recurrent and ∅ ≠ W(T) ∩ Tr(T) ⫋ W * (T) ∩ Tr(T) ⫋ QW(T) ∩ Tr(T) ⫋ BR(T) ∩ Tr(T), in which W * (T) is a recurrent points set related to an open question posed by Zhou and Feng. Specifically the set Tr(T) ∩ W * (T) \ W(T) is residual in X. Moreover, we construct a point x ∈ BR \ QW in symbol dynamical system, and demonstrate that the sets W(T),QW(T) and BR(T) of a dynamical system are all Borel sets.
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Supported by National Natural Science Foundation of China, Tian Yuan Special Foundation (Grant No. 11426198) and the Natural Science Foundation of Guangdong Province, China (Grant No. 2015A030310166)
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Wang, X.Y., Huang, Y. Recurrence of Transitive Points in Dynamical Systems with the Specification Property. Acta. Math. Sin.-English Ser. 34, 1879–1891 (2018). https://doi.org/10.1007/s10114-018-7534-7
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DOI: https://doi.org/10.1007/s10114-018-7534-7