A 9-branched generalized-Collatz example of G. Venturini
Consider the function T:
\[
T(x)=\left\{\begin{array}{ccc}
9x+1 &\mbox{if $x ≡ 0 \pmod{9}$}\\
\frac{x+32}{3} &\mbox{if $x ≡ 1 \pmod{9}$}\\
\frac{x-2}{3} &\mbox{if $x ≡ 2 \pmod{9}$}\\
x+3 &\mbox{if $x ≡ 3 \pmod{9}$}\\
\frac{100x-364}{9} &\mbox{if $x ≡ 4 \pmod{9}$}\\
\frac{x-5}{3} &\mbox{if $x ≡ 5 \pmod{9}$}\\
x-6 &\mbox{if $x ≡ 6 \pmod{9}$}\\
\frac{100x-637}{9} &\mbox{if $x ≡ 7 \pmod{9}$}\\
\frac{x-8}{3} &\mbox{if $x ≡ 8 \pmod{9}$}.
\end{array}
\right.
\]
This is Example 4. p. 208 of G. Venturini, Iterates of number-theoretic functions with periodic rational coefficients (generalization of the 3x+1 problem), Stud. Appl. Math. 86 (1992), no.3, 185-218.
It appears that all trajectories will eventually enter cycles 7 → 7, 4 → 4, or a cycle of the form 9t → 81t+1 → 27t+11 → 9t+3 → 9t+6 → 9t.
See link.
Last modified 29th June 2023
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