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Define the map $T_1$ on $\mathbb{F}_2[x]$ by $T_1(f)=\frac{f}{x}$ if $f(0)=0$and $T_1(f)=\frac{(x+1)f+1}{x}$ if $f(0)=1$. For a non-zero polynomial $f$ let$\tau_1(f)$ denote the least natural $k$ number for which $T_1^{k}(f)=1$.Define the average stopping time to be $\rho_1(n)=\frac{\sum_{f\in\mathbb{F}_2[x], \text{deg}(f)=n }\tau_1(f)}{2^n}$. We show that$\frac{\rho_1(n)}{n}$ converges to $2$ as $n\rightarrow\infty$ confirming aconjecture of Alon, Behajaina and Paran. Furthermore, we give a new proof that$\tau_1(f)\in O(\text{deg}(f)^{1.5})$ for all$f\in\mathbb{F}_2[x]\setminus\{0\}$.

On the average stopping time of the Collatz map in $\mathbb{F}_2[x]$