### Gonçalves - Greenfield - Madrid 4-branched generalized 3x+1 conjecture

We consider the d-branched mapping $$T: \mathbb{Z}\to\mathbb{Z}$$ given by $$\displaystyle T(x) = \left\lfloor\frac{m_ix}{d}\right\rfloor + x_i, \mbox{ if x \equiv i \pmod d},$$
with $$m_0=1, m_1=6, m_2=6, m_3=6; x_0=0, x_1 = 7, x_2=5, x_3=5, d=4$$:
$T(x)= \left\{ \begin{array}{cl} \frac{x}{4} & \mbox{ if x\equiv 0\pmod{4}}\\ \frac{3x+13}{2} & \mbox{ if x\equiv 1\pmod{4}}\\ \frac{3x+10}{2} & \mbox{ if x\equiv 2\pmod{4}}\\ \frac{3x+9}{2} & \mbox{ if x\equiv 3\pmod{4}} \end{array} \right.$ The iterates $$x, T(X), T(T(x)),\ldots$$ of the mapping are conjectured to eventually reach one of the cycles
• 0;
• 38, 62, 98, 152;
• 2, 8;
• 119, 183, 279, 423, 639, 963, 1449, 2180, 545, 824, 206, 314, 476;
• -9;
• -10;
• -43, -58, -82, -118, -172.
(This mapping is equivalent to the one on page 32 of the paper https://arxiv.org/pdf/2111.06170.pdf.)

The mapping can be regarded as a 4-branched example of type (b).

The associated Markov matrix $Q(4)= \left[ \begin{array}{cccc} 1/4 & 1/4 & 1/4 & 1/4\\ 1/2 & 0 & 1/2 & 0\\ 1/2 & 0 & 1/2 & 0\\ 0 & 1/2 & 0 & 1/2 \end{array} \right]$ has stationary vector $$(1/3, 1/6, 1/3, 1/6)$$ and we have the inequality

$(1/4)^{1/3}(3/2)^{1/6}(3/2)^{1/3}(3/2)^{1/6}< 1,$ thereby predicting everywhere eventual cycling.

Enter x (≠ 0):