Shift dynamical systems over finite fields

Abstract:

A trajectory over the finite field ${F_q}$ is a function from the integers I to ${F_q}$. The set $X({F_q})$ of all trajectories over ${F_q}$ is a topological vector space in the product topology induced by the discrete topology on ${F_q}$, and coordinatewise addition and scalar multiplication of trajectories. Let $\phi$ be a continuous linear operator on $X({F_q})$ which commutes with the shift. If x is a trajectory over ${F_q}$, then the $\phi$-orbit of x is the sequence of trajectories $x,\phi (x),{\phi ^2}(x), \cdots$. Suppose that $\phi$ is not a scalar multiple of the identity. Theorem. The trajectory x is periodic if and only if the $\phi$-orbit of x is eventually periodic.