论文标题
Koopman运营商和$ 3X+1 $ dyanical System
Koopman operators and the $3x+1$-dynamical system
论文作者
论文摘要
$ 3X+1 $ - 问题（或Collatz问题）是算术中臭名昭著的猜想。可以将其视为迭代地图，因此，它是自然数的离散空间$ \ mathbb {n} $上的动态系统。在当前的工作中研究了新兴的动力系统，该研究的方法是Koopman运营商和$ C^*$ - 代数的方法。这种方法使我们能够从状态空间（即集合$ \ mathbb {n} $）从状态空间中“提起” $ 3X+1 $ dyanical系统，即在状态空间上定义的功能空间，即序列空间。该提升的优点是可以通过有界线性操作员来描述Collatz问题，该线性算子构成了广泛研究的分析领域。我们研究这些运营商的特性及其与$ 3X+1美元的问题的关系。此外，我们使用傅立叶变换技术来研究从Collatz地图的轨迹中出现的符号序列的频率含量。这使我们能够在希尔伯特空间上定义一个等轴测图。最后，我们利用该等轴测图生成的$ C^*$ - 代数，以研究符号序列如何相互关联。
The $3x+1$-problem (or Collatz problem) is a notorious conjecture in arithmetic. It can be viewed as iterating a map and, therefore, it is a dynamical system on the discrete space $\mathbb{N}$ of natural numbers. The emerging dynamical system is studied in the present work with methods from the theory of Koopman operators and $C^*$-algebras. This approach enables us to "lift" the $3x+1$-dynamical system from the state space (i.e the set $\mathbb{N}$) to spaces of functions defined on the state space, i.e. to sequence spaces. The advantage of this lifting is that the Collatz problem can be described via bounded linear operators, which consist an extensively studied area of Analysis. We study the properties of these operators and their relationship to the $3x+1$-problem. Furthermore, we use Fourier transform techniques to investigate the frequency content of the sequences of signs emerging from the trajectories of the Collatz map. This enables us to define an isometry on a Hilbert space. Finally, we utilize the $C^*$-algebra generated by this isometry in order to study how the sequences of signs correlate with each other.
