Some patterns

Let us analyze the data of the following table, which displays the cycles of \(f_{D}\), for \(1 \leq D \leq 6\).

\(D\) Periods Cycles in \(N_{D}\)
\(1\) \(1\) \(\{0\}\)
\(2\) \(1\)
\(5\)
\(\{00\}\)
\(\{09, 81, 63, 27, 45\}\)
\(3\) \(1\)
\(5\)
\(\{000\}\)
\(\{099, 891, 693, 297, 495\}\)
\(4\) \(1\)
\(2\)

\(5\)
 
\(\{0000\}\)
\(\{2178, 6534\}\)
\(\{0999, 8991, 6993, 2997, 4995\}\)
\(\{0090, 0810, 0630, 0270, 0450\}\)
\(\{0909, 8181, 6363, 2727, 4545\}\)
\(5\) \(1\)
\(2\)

\(5\)
 
\(\{00000\}\)
\(\{21978, 65934\}\)
\(\{09999, 89991, 69993, 29997, 49995\}\)
\(\{09009, 81081, 63063, 27027, 45045\}\)
\(\{00990, 08910, 06930, 02970, 04950\}\)
\(6\) \(1\)

\(2\)




\(5\)




\(9\)

\(18\)

 
\(\{000000\}\)
\(\{219978, 659934\}\)
\(\{021780, 065340\}\)
\(\{099999, 899991, 699993, 299997, 499995\}\)
\(\{090009, 810081, 630063, 270027, 450045\}\)
\(\{009990, 089910, 069930, 029970, 049950\}\)
\(\{000900, 008100, 006300, 002700, 004500\}\)
\(\{009090, 081810, 063630, 027270, 045450\}\)
\(\{090909, 818181, 636363, 272727, 454545\}\)
\(\{099099, 891891, 693693, 297297, 495495\}\)
\(\{978021, 857142, 615384, 131868, 736263, 373626,\)
\(252747, 494505, 010989\}\)
\(\{043659, 912681, 726462, 461835, 076329, 847341,\)
\(703593, 308286, 374517, 340956, 318087, 462726,\)
\(164538, 670923, 341847, 406296, 286308, 517374\}\)

We see here some patterns. For each natural \(D\), the application \(f_{D}\) has only one fixed point. If \( D > 1\) is odd, then the cycles of \(N_{D}\) originate from the cycles of \(N_{D-1}\) by one of the following ways:

These procedures do not alter the period of the cycle, but not all possibilities produce cycles (for example, \(09909\) does not belong to any cycle of \(N_{5}\)).

See proof