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Non-wandering points and the depth for graph maps. (English) Zbl 1136.37007

Summary: Let \(G\) be a graph and \(f : G \rightarrow G\) be continuous. Denote by \(R (f)\) and \(\Omega (f)\) the set of recurrent points and the set of nonwandering points of \(f\), respectively. Let \(\Omega_{0}(f) = G\) and \(\Omega_{n}(f) = \Omega (f|_{\Omega_{n - 1} (f)})\) for all \(n \in \mathbb N\). The minimal \(m \in \mathbb N \cup \{\infty\}\) such that \(\Omega_{m}(f) = \Omega_{m +1}(f)\) is called the depth of \(f\).
In this paper, we show that \(\Omega_2 (f) = \overline {R(f)}\) and the depth of \(f\) is at most \(2\). Furthermore, we obtain some properties of nonwandering points of \(f\).

MSC:

37B10 Symbolic dynamics
54H20 Topological dynamics (MSC2010)
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References:

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