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 (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):

Last modified 8th March 2023
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