On attractors in one discrete binary dynamic system with bipartite dependency graph

One discrete binary dynamic system $(S_n,f)$, $n>1$, with bipartite dependency graph is considered. The states of such a system are all possible binary vectors of length $n$, and evolutionary function is $f=(x_n,0,\dots,0,x_1)$. In this case, $f$ is associated with a bipartite directed dependency graph with vertices set $\{a_1,\ldots,a_n,\epsilon\}$ and with arcs from $a_1$ to $a_n$, from $a_n$ to $a_1$ and from $a_i$ to $\epsilon$, $1<i<n$. The map of the $(S_3,f)$ system with the evolutionary function $f=(x_3,0,x_1)$ and its bipartite dependency graph are presented. A theorem is given on the type and number of attractors in these systems. Namely, the system has two attractors of length $1$: $0^n$ and $10^{n-2}1$, and one attractor of length $2$ formed by states $00^{n-2}1$ and $10^{n-2}0$.