On the average stopping time of the Collatz map in $\mathbb{F}_2[x]$
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909737206939648 |
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| author | Inselmann, Manuel |
| author_facet | Inselmann, Manuel |
| contents | 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 $τ_1(f)$ denote the least natural $k$ number for which $T_1^{k}(f)=1$. Define the average stopping time to be $ρ_1(n)=\frac{\sum_{f\in \mathbb{F}_2[x], \text{deg}(f)=n }τ_1(f)}{2^n}$. We show that $\lim_{n\rightarrow\infty}\frac{ρ_1(n)}{n}=2$, confirming a conjecture of Alon, Behajaina, and Paran. Furthermore, we give a new proof that $τ_1(f)\in O(\text{deg}(f)^{1.5})$ for all $f\in\mathbb{F}_2[x]\setminus\{0\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_12781 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the average stopping time of the Collatz map in $\mathbb{F}_2[x]$ Inselmann, Manuel Dynamical Systems Combinatorics Probability 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 $τ_1(f)$ denote the least natural $k$ number for which $T_1^{k}(f)=1$. Define the average stopping time to be $ρ_1(n)=\frac{\sum_{f\in \mathbb{F}_2[x], \text{deg}(f)=n }τ_1(f)}{2^n}$. We show that $\lim_{n\rightarrow\infty}\frac{ρ_1(n)}{n}=2$, confirming a conjecture of Alon, Behajaina, and Paran. Furthermore, we give a new proof that $τ_1(f)\in O(\text{deg}(f)^{1.5})$ for all $f\in\mathbb{F}_2[x]\setminus\{0\}$. |
| title | On the average stopping time of the Collatz map in $\mathbb{F}_2[x]$ |
| topic | Dynamical Systems Combinatorics Probability |
| url | https://arxiv.org/abs/2401.12781 |