Orbit of shift operator in symbolic dynamics

Question

Consider the symbolic space of infinite $0-1$ words, $\{0,1\}^{\mathbb{N}}$. This can be shown to be a compact metric space with $$d(s, \bar{s}):=\displaystyle\sum_{t = 1}^{\infty}\frac{\delta_t}{2^{|t|}}, \text{ where } \delta_t:=\begin{cases}0 & \text{if } s_t=\bar{s}_t \\ 1 & \text{if } s_t\neq\bar{s}_t\end{cases}.$$ Now, consider the (non-invertible) shift function $\sigma:\{0,1\}^{\mathbb{N}}\to\{0,1\}^{\mathbb{N}}$ defined as $\sigma(x(n)) = x(n+1)$. I know that $\sigma$ is continuous, turning $(\{0,1\}^{\mathbb{N}}, \sigma)$ into a natural dynamical system. Now, my question is: which words in this symbolic space have dense orbit under $\sigma$? My suspicion is that it has to do with every possible finite $0-1$ word appearing in the sequence, but I'm not sure how to prove it.

Answer 1

Indeed, your suspicion is correct. Let $w\in\{0,1\}^{\mathbb{N}}$, and let $\overline{w}=\overline{\{\sigma^n w\}}$, the orbit closure of $\sigma$ on $w$. We claim that the orbit of $\sigma$ on $w$ is dense, i.e. $\overline{w}=\{0,1\}^{\mathbb{N}}$ $\iff$ any finite $0-1$ word appears as a subsequence of $w$.

$(\Leftarrow)$ Let $w'\in\{0,1\}^{\mathbb{N}}$ and $\varepsilon>0$. Moreover, choose $N\in\mathbb{N}$ such that $2^{-N}<\varepsilon$. The subsequence $w'_1w'_2\ldots w'_N$ appears in $w$, say at some starting index $i$. Then, $d(\sigma^i w,w')\le 2^{-N}<\varepsilon,$ as wanted.

$(\Rightarrow)$ Note that for some $N\in\mathbb{N}$, $d(w, w')<2^{-N}$ only if $w_1w_2\ldots w_N = w'_1w'_2\ldots w'_N$. Since the orbit $\sigma^nw$ is dense, for any $w'\in\{0,1\}^{\mathbb{N}}$ we can choose some integer $i$ such that $d(\sigma^iw,w')<2^{-N}$, which means $w'_1w'_2\ldots w'_N$ must occur as a subsequence of $w$. The choice of $w'$ was arbitrary, so we are done.

As an aside, $\sigma$ becomes invertible when you consider it as a continuous map on $\{0,1\}^{\mathbb{Z}}$, the space of bilaterally infinite words, which can be turned into a compact metric space similarly (and is thus a topological dynamical system in the same sense).