Let $G$ be finitely generated group with $G=\langle S\rangle$ and $\{0, 1\}^G$ endows with metric $d:\{0, 1\}^G\times \{0, 1\}^G\to \mathbb{R}$ define by \begin{equation*} d(x, y)= 2^{-\inf\{|g|_S: g\in G, x_g\neq y_g\}}, \end{equation*} and $|g|_S$ is a word metric associated to a finite set $S$ of generators of $G$. We consider the action $\sigma:G\times \{0, 1\}^G\to \{0, 1\}^G$ of $G$ on $\{0, 1\}^G$ by right shifts, i.e. $\sigma(g, x)_h = x_{hg}$ for every $g,h\in G$ and $x\in\{0, 1\}^G$.

It is known that if $G=\mathbb{Z}$, then $\sigma:\{0, 1\}^\mathbb{Z}\to \{0, 1\}^\mathbb{Z}$ has a dense orbit and the set of periodic points of $\sigma$ is dense in $\{0, 1\}^\mathbb{Z}$.

I would like to know that shift action $\sigma:G\times \{0, 1\}^G\to \{0, 1\}^G$ has a dense orbit or not?

Can you please help me or introduce a book or a paper in this subject?


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