论文标题
锡拉丘兹地图作为非符号功率结合的变换及其逆图
Syracuse Maps as Non-singular Power-Bounded Transformations and Their Inverse Maps
论文作者
论文摘要
我们证明了动态系统$（\ MATHBB {n}，2^{\ MATHBB {n}}，T，μ）$，其中$μ$是等同于计数量度等同的有限度量，在$ l^1（μ）$中均为$ l^1（μ）$，并且在其中存在一个$ x $ to $ x $ a $ x $ x \ in of the $ x（μ） $ k \ in \ mathbb {n} $，这样$ t^k（x）$在地图$ t $的某个周期中。该结果对Collatz的猜想具有直接的影响，我们使用它来激励研究逆映像的数字理论属性$ t^{ - 1}（x）$ for $ x \ in \ mathbb {n} $，其中$ t $在其中表示Collatz Map。我们研究了相关锡拉丘兹地图的相似特性，并将其与Collatz地图进行了比较。我们还分析了反图像的某些结构属性，相对于集合的渐近密度，$ \ {x \ in \ mathbb {n} \ mid \ in \ in \ mathbb {n}中的k \ in \ mathbb {n}：t^k（x）<x \ \ \} $。
We prove that the dynamical system $(\mathbb{N}, 2^{\mathbb{N}}, T, μ)$, where $μ$ is a finite measure equivalent to the counting measure, is power-bounded in $L^1(μ)$ if and only if there exists one cycle of the map $T$ and for any $x \in \mathbb{N}$, there exists $k \in \mathbb{N}$ such that $T^k(x)$ is in some cycle of the map $T$. This result has immediate implications for the Collatz Conjecture, and we use it to motivate the study of number theoretic properties of the inverse image $T^{-1}(x)$ for $x \in \mathbb{N}$, where $T$ denotes the Collatz map here. We study similar properties for the related Syracuse maps, comparing them to the Collatz map. We also analyze some structural properties of the inverse image in relation to asymptotic density of the set $\{x \in \mathbb{N} \mid \exists k \in \mathbb{N}: T^k(x) < x\}$.
