$(4,1/2)$ reversible cellular automaton, rule BB991133

This automaton has Welch indices $L=2$ and $R=2$. The evolution rule, an example of the evolution, and its block permutations are the following:

Figure 44: Evolution of the $(4,1/2)$ reversible one dimensional cellular automaton rule $BB991133$

The connectivity relation associated with this automaton is the following:

Figure 45: Connectivity relation of the $(4,1/2)$ reversible one dimensional cellular automaton rule $BB991133$

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

Figure 46: Transitive closure of the connectivity relation of the $(4,1/2)$ reversible one dimensional cellular automaton rule $BB991133$

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

Figure 47: Classes of the transitive closure of the connectivity relation of the $(4,1/2)$ automaton rule $BB991133$

In this case we obtain the same equivalence class that in the example of section 7.3, therefore both automata belongs to the same dynamical class. Take the block $5,7$ 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 48: 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 49: 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 50: 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 51: 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 ${\mathcal{C}_{[303]}}$ to the centered cylinder set ${\mathcal{C}_{[111]}}$ in $6$ steps, corresponding to $12$ evolutions because the composition of the evolution rule. We use the recurrent centered cylinder set ${\mathcal{C}_{[131]}}$ for constructing such an orbit.

Figure: Orbit from the centered cylinder set ${\mathcal{C}_{[303]}}$ to the centered cylinder set ${\mathcal{C}_{[111]}}$ in $6$ steps

Since the centered cylinder set ${\mathcal{C}_{[131]}}$ can be fixed, we can use it to get an orbit from the centered cylinder set ${\mathcal{C}_{[303]}}$ to the centered cylinder set ${\mathcal{C}_{[111]}}$ in $7$ steps.

Figure: Orbit from the centered cylinder set ${\mathcal{C}_{[303]}}$ to the centered cylinder set ${\mathcal{C}_{[111]}}$ in $7$ steps