I'm currently investigating Wolfram's elementary cellular automata on finite grids with periodic boundary conditions, i.e. on $\mathbb{Z}/k$ for different $k$.

It is clear that for each rule $R$ and each state space $\Sigma_k = \{2^k\}$ there are three kinds of states:

garden eden states $\alpha$, i.e. states without predecessor $\beta$ with $\Phi_R(\beta) = \alpha$

periodic states $\omega$, i.e. states lying on a limit cycle

transient states $\tau$, i.e. states lying on shortest trajectories going from some $\alpha$ to some $\omega$.

It turns out that there are limit cycles of which some but not all periodic states $\omega$ are "targets" of some garden eden state: their unique predecessor lies on the same limit cycle. Let's call these states $\gamma$ *pseudo-garden eden states* (without further reason).

I wonder if $\gamma$ states were found worth to be investigated. We know that garden eden states may be characterized by local patterns not reachable/producable by any predecessor $\beta$. **Might there be a similar characterization of pseudo-garden eden states?**

For some rules $R$ and numbers $k$ pseudo-garden eden states appear periodically, for others they come irregularly. In the following examples the **non**-$\gamma$ states – i.e. states where a transient meets the limit cycle – are marked with their index.

**Rule 26**

**Rule 73**

**Rule 110**

**Rule 18**

**Rule 45**