Given a finite set $F$ with the discrete topology, $F^{Z}$ with the product topology is homeomorphic to the Cantor set. In symbolic dynamics we study iterations of the shift operator: $\sigma: F^{Z}\rightarrow F^{Z}$ defined by $\sigma(x)_{i} = x_{i+1}$. This talk focuses on the relationship between the shift on $(Z/2Z)^{Z}$ and another important continuous map $\tau : (Z/2Z)^{Z}\rightarrow (Z/2Z)^{Z}$ defined by $\tau(x)_{i} = x_{i} + x_{i+1}$. In particular, we describe a class of closed subsets of $(Z/2Z)^{Z}$ which are invariant under $\tau$ and $\sigma$.