Let $X=\{0,1\}^{\mathbb{N}}$ and $\xi_n$ be the partition of $X$ defined by the equivalence relation $x \sim_n x' \Leftrightarrow (x_{n}, x_{n+1}, \ldots) = (x_{n}', x_{n+1}', \ldots)$. The sequence of partitions $(\xi_n)$ is decreasing and we introduce the intersection partition $\theta= \cap \xi_n$. I'm loooking for a group $G$ of transformations of $X$ such that $\theta$ is the partition into the orbits of $G$.

Maybe my question is somewhat unprecise, I am rather new in ergodic theory. Any comments are welcomed.