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.
Last modified 8th March 2023
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