## 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 $$f_{D,2}$$).

See proof

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 $$\left(\frac{B}{3},\frac{B}{3}-1,\frac{2B}{3}-1,\frac{2B}{3}\right)_{B}$$.

See proof