Some patterns II

For \(D\) even, it seems there are more ways of building the cycles of \(N_{D}\). For example, we can fix an element of a cycle of \(N_{D-2}\) and place a zero in each extremity (such as in \(\{0090, 0810, 0630, 0270, 0450\}\) when \(D=4\)); or choose an element of a cycle of \(N_{\frac{D}{2}}\) and repeat it twice (as in \(\{0909, 8181, 6363, 2727, 4545\}\), when \(D=4\)); or select cycles for smaller values of \(D\) and join them after a suitable permutation (such as in the cycle \[\{978021, 857142, 615384, 131868, 736263, 373626, 252747, 494505, 010989\}\] of \(D = 6 = 2+4\), whose first element comes from joining in this way the cycle \(\{09, 81, 63, 27, 45\}\) of \(N_{2}\) and the cycle \(\{2178, 6534\}\) of \(N_{4}\)).

The list of procedures detected for \(1 \leq D \leq 12\) is a large one, with some of them generating cycles with periods which are new in relation to those already obtained for smaller values of \(D\). The algorithm elaborated by Atractor to compute the cycles of \(f_{D}\) took hundredths of a second to give the complete list of such special orbits when \(2 \leq D \leq 4\); it took some few seconds for \(D = 6\) and about two minutes for \(D = 8\). However, for \(D = 12\), the expected time rose to about one year (although, using random samples of elements in the image of \(f_{12}\), the search for cycles became faster). This leaves the challenge of proving, for \(D =12\) and without the help of a computer, some of the dynamical properties previously detected.