$(4,1/2)$ reversible cellular automaton, rule AA5500FF

This automaton has Welch indices $L=1$ and $R=4$. The evolution rule, an example of the evolution, and its block permutations are the following:

Figure 34: Evolution of the $(4,1/2)$ reversible one dimensional cellular automaton rule $AA5500FF$

The connectivity relation associated with this automaton is the following:

Figure 35: Connectivity relation of the $(4,1/2)$ reversible one dimensional cellular automaton rule $AA5500FF$

The transitive closure of connectivity relation associated with this automaton is the following:

Figure 36: Transitive closure of the connectivity relation of the $(4,1/2)$ reversible one dimensional cellular automaton rule $AA5500FF$

We have the following classes:

Figure 37: Classes of the transitive closure of the connectivity relation of the $(4,1/2)$ automaton rule $AA5500FF$

In this case we have $10$ classes of $6$ elements and $2$ classes of $2$ elements. Take the block $0,11$ representing the sequence of states $102$. This block has period $6$, so the configuration formed with repetitions of the sequence $102$ must have period $6$, or period $12$ in the evolution of the automaton.

An example of this periodical behavior is the following:

Figure 38: Period $12$ corresponding to a period $6$ using the composition of the evolution rule in the initial configuration formed with repetitions of the sequence $102$

Now, we will see all the possible mappings among sequences of $3$ cells using the process described in section 5.4. For example, the mapping of $203$ is the following:

Figure 39: All the possible mappings from the sequence $203$

Calculating all the mappings among sequences of $3$ cells, we have the following mapping among centered cylinder sets:

Figure 40: Mapping among centered cylinder sets, the dark points indicate recurrent centered cylinder sets

The transitive closure of the mapping among centered cylinder sets is the following:

Figure 41: Transitive closure of the mapping among centered cylinder sets, the dark points indicate recurrent centered cylinder sets

Since we only have one equivalence class and there exists centered cylinder sets that can be fixed, then this automaton has topologically mixing orbits. For example, we can form an orbit from the centered cylinder set ${\mathcal{C}_{[011]}}$ to the centered cylinder set ${\mathcal{C}_{[030]}}$ in $6$ steps, corresponding to $12$ evolutions because the composition of the evolution rule. We use the recurrent centered cylinder set ${\mathcal{C}_{[020]}}$ for constructing such an orbit.

Figure: Orbit from the centered cylinder set ${\mathcal{C}_{[011]}}$ to the centered cylinder set ${\mathcal{C}_{[030]}}$ in $6$ steps

Since the centered cylinder set ${\mathcal{C}_{[020]}}$ can be fixed, we can use it to get an orbit from the centered cylinder set ${\mathcal{C}_{[011]}}$ to the centered cylinder set ${\mathcal{C}_{[030]}}$ in $7$ steps.

Figure: Orbit from the centered cylinder set ${\mathcal{C}_{[011]}}$ to the centered cylinder set ${\mathcal{C}_{[030]}}$ in $7$ steps