Is anyone familiar with results about properties of the limit set of the local rule for a cellular automaton? I haven't been able to find any good materials on the subject from an initial search, and I thought that someone might help refer me to such results. I recently asked the same question on a thread in math stack exchange, but maybe someone here will have a more helpful suggestion.

I am considering a cellular automaton (CA) $G$ on $\mathcal{A}^{\mathbb{Z}^d}$ with a local rule given by a local rule $f:\mathcal{A}^N\to \mathcal{A}$, where $N$ is some memory neighborhood and $\mathcal{A}$ is the finite set of states. If we denote the limit set $\Omega_G= \cap G^n \big( \mathcal{A}^{\mathbb{Z}^d}\big)$, which is a subshift, what properties can be said on $\Omega_G$ as a function of $f$? I am interesting in the following questions:

- Is $\Omega_G$ a transitive\minimal , with respect to the shift action, if some conditions on $f$ hold?
- Are there conditions on $f$ such that $\Omega_G$ is recurrent or linearly recurrent?

I know that there are implications of these sort if we talk instead on substitution and their limit set, but I do not know if there are similar statements for CA. I have been looking at these notes, by Jarkko Kari on his website, to try to see whether there are such statements but I haven't found any there. I also searched shortly in the book Cellular automata and groups and have not found anything productive.

Is anyone familiar with works in this direction?