$(4,1/2)$ reversible cellular automaton, rule FFAA5500

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 14: Evolution of the $(4,1/2)$ reversible one dimensional cellular automaton rule $FFAA5500$

The connectivity relation associated with this automaton is the following:

Figure 15: Connectivity relation of the $(4,1/2)$ reversible one dimensional cellular automaton rule $FFAA5500$, the dark points show fixed configurations

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

Figure 16: Transitive closure of the connectivity relation of the $(4,1/2)$ reversible one dimensional cellular automaton rule $FFAA5500$

If we rearrange this transitive closure, we obtain the following classes:

Figure 17: Rearrange of the transitive closure of the connectivity relation of the $(4,1/2)$ automaton rule $FFAA5500$

In this case we see that the automaton have $24$ equivalence classes, $20$ of $3$ elements each one and $4$ of one element. Take the block $1,8$ representing the sequence of states $102$. This block has period $3$, so the configuration formed with repetitions of the sequence $102$ must have period $3$. We have to remember that we are using the composition of the original evolution rule for keeping the same position when we compare configurations. In this way, the period $3$ is truly a period $6$ in the evolution of the automaton. An example of this periodical behavior is the following:

Figure 18: Period $6$ corresponding to a period $3$ 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 19: All the possible mappings from the sequence $203$

Calculating all the possible mappings among sequences of $3$ cells, and taking such sequences as centered cylinder sets, we have the following mapping among centered cylinder sets:

Figure 20: 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 21: 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 ${\mathcal{C}_{[203]}}$ to the centered cylinder set ${\mathcal{C}_{[312]}}$ in $6$ steps, corresponding to $12$ evolutions because the composition of the evolution rule. We use the recurrent centered cylinder set ${\mathcal{C}_{[222]}}$ for constructing such an orbit.

Figure: Orbit from the centered cylinder set ${\mathcal{C}_{[203]}}$ to the centered cylinder set ${\mathcal{C}_{[312]}}$ in $6$ steps

But, since the centered cylinder set ${\mathcal{C}_{[222]}}$ can be fixed, we can use it to get an orbit from the centered cylinder set ${\mathcal{C}_{[203]}}$ to the centered cylinder set ${\mathcal{C}_{[312]}}$ in $7$ steps.

Figure: Orbit from the centered cylinder set ${\mathcal{C}_{[203]}}$ to the centered cylinder set ${\mathcal{C}_{[312]}}$ in $7$ steps