Almost all orbits of an analogue of the Collatz map on the reals attain bounded values
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911767891804160 |
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| author | Inselmann, Manuel |
| author_facet | Inselmann, Manuel |
| contents | Motivated by a balanced ternary representation of the Collatz map we define the map $C_\mathbb{R}$ on the positive real numbers by setting $C_\mathbb{R}(x)=\frac{1}{2}x$ if $[x]$ is even and $C_\mathbb{R}(x)=\frac{3}{2}x$ if $[x]$ is odd, where $[x]$ is defined by $[x]\in\mathbb{Z}$ and $x-[x]\in(-\frac{1}{2},\frac{1}{2}]$. We show that there exists a constant $K>0$ such that the set of $x$ fulfilling $\liminf_{n\in\mathbb{N}}C_\mathbb{R}^n(x)\leq K$ is Lebesgue-co-null. We also show that for any $ε>0$ the set of $x$ for which $ (\frac{3^{\frac{1}{2}}}{2})^kx^{1-ε}\leq C_\mathbb{R}^k(x)\leq (\frac{3^{\frac{1}{2}}}{2})^kx^{1+ε}$ for all $0\leq k\leq \frac{1}{1-\frac{\log_23}{2}}\log_2x$ is large for a suitable notion of largeness. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_17241 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Almost all orbits of an analogue of the Collatz map on the reals attain bounded values Inselmann, Manuel Dynamical Systems Combinatorics Number Theory Probability Motivated by a balanced ternary representation of the Collatz map we define the map $C_\mathbb{R}$ on the positive real numbers by setting $C_\mathbb{R}(x)=\frac{1}{2}x$ if $[x]$ is even and $C_\mathbb{R}(x)=\frac{3}{2}x$ if $[x]$ is odd, where $[x]$ is defined by $[x]\in\mathbb{Z}$ and $x-[x]\in(-\frac{1}{2},\frac{1}{2}]$. We show that there exists a constant $K>0$ such that the set of $x$ fulfilling $\liminf_{n\in\mathbb{N}}C_\mathbb{R}^n(x)\leq K$ is Lebesgue-co-null. We also show that for any $ε>0$ the set of $x$ for which $ (\frac{3^{\frac{1}{2}}}{2})^kx^{1-ε}\leq C_\mathbb{R}^k(x)\leq (\frac{3^{\frac{1}{2}}}{2})^kx^{1+ε}$ for all $0\leq k\leq \frac{1}{1-\frac{\log_23}{2}}\log_2x$ is large for a suitable notion of largeness. |
| title | Almost all orbits of an analogue of the Collatz map on the reals attain bounded values |
| topic | Dynamical Systems Combinatorics Number Theory Probability |
| url | https://arxiv.org/abs/2401.17241 |