Transitive behavior of $({k^{}},1/2)$ reversible one dimensional cellular automata

We have seen that some restrictions in the configurations define both periodic orbits and non-wandering centered cylinder sets in $({k^{}},1/2)$ reversible one dimensional cellular automata. Now is desirable to consider not only one but all the possible mappings that a sequence ${w_{}}\in {K^{3}}$ has to. This can be done using again block permutations and the process is the following:

1. For a sequence ${w_{i}}$ of states in the set ${K^{3}}$, take its mapping to an unique block $x_iy_i$ using the block permutation $p_1$.
2. Associate the element $x_i$ with all the elements $y$ in the set $Y$ and the element $y_i$ with all the elements $x$ in the set $X$.
3. With the associations $yx_i$ and $y_ix$ and using the block permutation $p_2^{-1}$, form the respective list of the mappings from these associations to sequences ${w_{j}}$ and ${w_{k}}$ of states in the set ${K^{3}}$.
4. Using the block permutation $p_1$, every list of sequences ${w_{j}}$ and ${w_{k}}$ of states defined by $yx_i$ and $y_ix$, maps respectively to a list of blocks with the form $x_jy_j$ and $x_ky_k$.
5. From the last lists, take only all the different elements $y_j$ in the first list and all the different elements $x_k$ in the second list.
6. Form the cartesian product of such lists. This cartesian product form a new list of blocks with the form $y_jx_k$.
7. The list of blocks $y_jx_k$ maps to a list of sequences ${w_{m}}$ using the block permutation $p_2^{-1}$. Thus, we have a subset of the set ${K^{3}}$.

In this way, we have that an initial sequence ${w_{i}}$ maps to a set of sequences ${w_{m}}$ placed in the same position and therefore we have all the possible mappings from sequences to sequences in the set ${K^{3}}$ as is showed in Figure 12. tr

Figure 12: Description of all the possible mappings from a sequence ${w_{i}}$ to a set of sequences ${w_{m}}$ using block permutations

Using all the sequences ${w_{}}$ of states in the set ${K^{3}}$ for defining centered cylinder sets ${\mathcal{C}_{[{w_{}}]}}$, the previous process defines the possible mappings from a centered cylinder set to other centered cylinder sets. In this way, we can use this process for detecting transitive behavior among centered cylinder sets that cover all the configuration space ${({{C}_{}},{\mathfrak{C}_{}})}$.