LAUSR

Let $$\mathbf{D}$$D be a dendrite with finite branch points and $$f:\mathbf{D}\rightarrow \mathbf{D}$$f:D→D be continuous. Denote by R(f) and $$\Omega (f)$$Ω(f) the set of recurrent points and the set of non-wandering points of f respectively. Let $$\Omega _0 (f)=\mathbf{D}$$Ω0(f)=D and $$\Omega _n (f)=\Omega (f|_{\Omega _{n-1} (f)})$$Ωn(f)=Ω(f|Ωn-1(f)) for all $$n\in \mathbf{N}$$n∈N. The minimal $$m\in \mathbf{N}\cup \{\infty \}$$m∈N∪{∞} such that $$\Omega _{m} (f)=\Omega _{m+1} (f)$$Ωm(f)=Ωm+1(f) is called the depth of f. In this note, we show that $$\Omega _3(f)=\overline{R(f)}$$Ω3(f)=R(f)¯ and the depth of f is at most 3. Furthermore, we show that there exist a dendrite $$\mathbf{D}$$D with finite branch points and $$f\in C^0(\mathbf{D})$$f∈C0(D) such that $$ \Omega _3(f)=\overline{R(f)}\ne \Omega _2(f)$$Ω3(f)=R(f)¯≠Ω2(f).