The dynamics of a trick

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 $$N_{2}$$.