Let \(\mathbf{D}\) be a dendrite with finite branch points and \(f:\mathbf{D}\rightarrow \mathbf{D}\) be continuous. Denote by R(f) and \(\Omega (f)\) the set of recurrent points and the set of non-wandering points of f respectively. Let \(\Omega _0 (f)=\mathbf{D}\) and \(\Omega _n (f)=\Omega (f|_{\Omega _{n-1} (f)})\) for all \(n\in \mathbf{N}\). The minimal \(m\in \mathbf{N}\cup \{\infty \}\) such that \(\Omega _{m} (f)=\Omega _{m+1} (f)\) is called the depth of f. In this note, we show that \(\Omega _3(f)=\overline{R(f)}\) and the depth of f is at most 3. Furthermore, we show that there exist a dendrite \(\mathbf{D}\) with finite branch points and \(f\in C^0(\mathbf{D})\) such that \( \Omega _3(f)=\overline{R(f)}\ne \Omega _2(f)\).

中文翻译：

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具有唯一分支点的树突上连续映射的中心和中心深度

令 \(\mathbf{D}\) 是一个具有有限分支点的树突，并且 \(f:\mathbf{D}\rightarrow \mathbf{D}\) 是连续的。分别用 R(f) 和 \(\Omega (f)\) 表示 f 的循环点集和非游荡点集。令 \(\Omega _0 (f)=\mathbf{D}\) 和 \(\Omega _n (f)=\Omega (f|_{\Omega _{n-1} (f)})\) 为所有 \(n\in \mathbf{N}\)。最小 \(m\in \mathbf{N}\cup \{\infty \}\) 使得 \(\Omega _{m} (f)=\Omega _{m+1} (f)\) 是称为 f 的深度。在本笔记中，我们证明 \(\Omega _3(f)=\overline{R(f)}\) 和 f 的深度最多为 3。此外，我们证明存在一个枝晶 \(\mathbf{ D}\) 与有限分支点和 \(f\in C^0(\mathbf{D})\) 使得 \( \Omega _3(f)=\overline{R(f)}\ne \Omega _2 （F）\）。