The dynamics of a trick 
Considering other bases?
The mapping \(f_{D}\) may been seen as acting on the set of natural numbers when represented in another base other than \(10\) (let us denote it by \(f_{D,base}\)). One expects distinct dynamics, since the behavior of the orbits of \(f_{D,base}\) depends on the digits allowed in the representation of the naturals. Click here to compare the behavior of the orbits of \(f_{7,2}\) and \(f_{5,3}\).
For example, for the base \(2\) and for \(D = 4\), the image of such a transformation \(f_{4,2}\) contains five natural numbers and there are four fixed cycles, namely \(\{0000\}\), \(\{0010\}\), \(\{0101\}\), \(\{0111\}\); and these are the only attractors of \(f_{4,2}\).
Cycles and precycles in \(f_{4,2}\).
More generally, in this base and whenever \(D\) is even, there are \(2^{\frac{D}{2}}\) fixed points (and there is a conjecture stating that these are the only attractors of \(f_{D,2}\)).
In base \(3\) and for \(D = 6\), we find three fixed points \((\{000000\}\), \(\{010120\}\), \(\{102212\})\) and an orbit of preperiod \(2\) \((\{010212, 201021\})\), and only these cycles.
Cycle of preperiod \(2\) of \(f_{6,3}\) with its preimages.
Click on the picture to see it in a bigger size.
One can prove that, for some \(D\), \(f_{D,B}\) has nonzero fixed points if and only if \(B\) is not congruent to \(1\) modulo \(3\). If \(B\) is congruent to \(2\) modulo \(3\) then \(\left(\frac{B-2}{3},\frac{2B-1}{3}\right)_{B}\) is a fixed point of \(f_{2,B}\); if \(B\) is a multiple of \(3\) then a fixed point of \(f_{4,B}\) is \(\left(\frac{B}{3},\frac{B}{3}-1,\frac{2B}{3}-1,\frac{2B}{3}\right)_{B}\).
