论文标题
Collatz地图作为非单一转换
Collatz map as a non-singular transformation
论文作者
论文摘要
令$ t $为$ \ n = \ {1,2,3，... \} $ by $ t（n）= \ frac {n} {2} $，如果$ n $均匀，$ t（n）= \ frac {3n+1} {2} {2} $ n $是ODD。考虑动态系统$（\ n，2^{\ n}，t，μ）$，其中$μ$是计数量度。此动态系统$（\ n，2^{\ n}，t，μ）$具有以下属性。 \ begin {enumerate} \项目存在一个不变的有限度量$γ$，因此所有$ a \ subset \ n. $γ（a）\leqμ（a）$（a）$（a \ subset \n。$ f \ in l^1（μ）中的每个函数$ f \ intem每$ x \ in \ n $ conver to $ f^*（x）$其中$ f^* \ in l^1（μ）。$ \ end end {Enumerate}，我们还表明，Collatz的猜想等于存在有限的度量$ c $ n，\ n，2^{\ n，\ n，2^{\ n} $ vf = t ocive t ocive the $ n， $ l^1（ν）$带有conerrvative零件$ \ {1,2 \}。$
Let $T$ be the map defined on $\N=\{1,2,3, ...\}$ by $T(n) = \frac{n}{2} $ if $n$ is even and by $T(n) = \frac{3n+1}{2}$ if $n$ is odd. Consider the dynamical system $(\N, 2^{\N}, T,μ)$ where $μ$ is the counting measure. This dynamical system $(\N, 2^{\N}, T, μ)$ has the following properties. \begin{enumerate} \item There exists an invariant finite measure $γ$ such that $γ(A) \leq μ(A) $ for all $A \subset \N.$ \item For each function $f\in L^1(μ)$ the averages $\frac{1}{N} \sum_{n=1}^N f(T^nx)$ converge for every $x\in \N$ to $ f^*(x)$ where $ f^* \in L^1(μ).$ \end{enumerate} We also show that the Collatz conjecture is equivalent to the existence of a finite measure $ν$ on $(\N, 2^{\N})$ making the operator $Vf = f\circ T$ power bounded in $L^1(ν)$ with conserrvative part $\{1,2\}.$
