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In this question there is an example of a continuous dynamical system with $NW(f) \not\subset \overline{R(f)}$. The definitions I am working with are exactly same as that question. I want to find a symbolic or a discrete dynamical system with the mentioned property. I would really appreciate a not-so-involved example, so that I can gain an intuition to distinguish non-wandering points from points in the closure of the set of all recurrent points.

テレビ スクリーン
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1 Answers1

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One example is the orbit closure of ${}^\infty 0 1 0 1 0 0 1 0 0 0 1 \cdots 1 0^n 1 0^{n+1} 1 \cdots$ with respect to the shift map. The left tail is just repeating 0s, and on the right are 1s with increasing gaps. The only recurrent point is the all-0 configuration, but ${}^\infty 0 1 0^\infty$ is nonwandering.

Ilkka Törmä
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  • There is no other element in the orbit closure than $\infty 010\infty$ and all-$0$ which is not in the orbit, right? Btw, thank you very much, I think this example fits perfectly to what I was searching. – テレビ スクリーン Mar 24 '21 at 19:23
  • @テレビ スクリーン Correct. The shift space contains the point I defined, the point ${}^\infty 010^\infty$, the all-0 point and their shifts, nothing more. – Ilkka Törmä Mar 24 '21 at 19:36