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Recurrence in pairs. (English) Zbl 1162.37005

The paper studies the notion of weak product recurrence. Classically, a topological dynamical system consist of a topological space \(x\) with a one-parameter group \(\{T^n\mid n\in\mathbb{Z} \}\) of homeomorphisms acting on \(x\). For a point \(x\in X\) and a neighborhood \(U\) of \(x\), the return time set is defined as
\[ R(x,U)=\{n\in\mathbb{Z}\mid T^n_x\in U\}. \]
In this setting, a point \(x\) is said to be weakly product recurrence if given any uniformly, recurrent point \(y\) in any dynamical system and any neighborhoods \(U\) of \(x\) and \(v\) of \(y\), the return time set \(R(x,U)\) and \(R(y,V)\) intersect non-trivially. The paper establishes two sufficient conditions for weak product recurrence. It is deduced that any point with a dense orbit in either the full one-sided shift on a finite number of symbols or a mixing subshift of finite type is weakly product recurrent. J. Auslander and H. Furstenberg [Trans. Am. Math. Soc. 343, No. 1, 221–232 (1994; Zbl 0801.54031)] asked if weak product recurrence implies distality. The paper answers this question in a negative way.

MSC:

37B10 Symbolic dynamics
37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
37B50 Multi-dimensional shifts of finite type, tiling dynamics (MSC2010)

Citations:

Zbl 0801.54031
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References:

[1] Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory (1981) · Zbl 0459.28023 · doi:10.1515/9781400855162
[2] van der Waerden, Nieuw. Arch. Wiskd. 15 pp 212– (1927)
[3] DOI: 10.2307/2154530 · Zbl 0801.54031 · doi:10.2307/2154530
[4] DOI: 10.1090/S0002-9947-00-02704-5 · Zbl 0976.54039 · doi:10.1090/S0002-9947-00-02704-5
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