What happens if we consider only one or two digits?

We could have started by considering a transformation \(f_{1}\) similar to \(f\) but acting on the set \(N_{1}\) of natural numbers with only one digit: in that case, \(f_{1}\) maps each number to the fixed point \(0\). On the set \(N_{2}\) of natural numbers with two digits, besides the fixed point \(00\), the transformation \(f_{2}\) has a cycle of period \(5\) \[09\rightarrow 81\rightarrow 6\rightarrow 27\rightarrow 45\] which is reached in one or two iterations of \(f_{2}\) by all numbers \(ab\) with \(a\neq b\). Observe that this cycle is related with the cycle of period \(5\) in \(N_{3}\): the cycle in \(N_{3}\) is obtained from the cycle in \(N_{2}\) by placing a \(9\) in the middle of each number. The following figure illustrates the dynamics of \(f_{2}\).

Cycles and precycles in \(N_{2}\).