# Shift spaces on groups

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?