Let's think about some finite field $\mathbb{F}$. Is it possible to construct a map
$x[n+1] = \mathcal{P}(x[n], x[n-1],...,x[n-k]), \ \ \ \forall x\in\mathbb{F} $
where $\mathcal{P}$ - polynomial with coeffs $\in\mathbb{F}$, so it's behavior be non-periodic, but chaotic,
so x[n] be "jumping" randomly over $\mathbb{F}$?
(As $x[n+1] =1-2\cdot x[n]^2$ on [-1,1])