The numerical hailstone sequences, or orbits, generated by the Collatz map have been disclosed to present relevant features commonly associated with complex systems. It is so despite the extreme simplicity of the arithmetic dynamical system iteration rule. Indeed, for a positive integer n, the Collatz map f reads f(n)=n/2 ( f(n)=3n+1 ) for n even (odd). Seeking to elucidate this surprising fact, here we unveil distinct characteristics of stochastic-like behavior for collections of Collatz orbits by considering methods commonly employed to temporal series, as cryptography tests, power-spectrum, detrended fluctuation, auto-correlation and entropy measure. Besides confirming previous predictions that the Collatz orbits display some global properties of geometric Brownian motion, our results are likewise able to explain, at least heuristically, the reasons for so. In special, we show by means of comprehensive analysis that our findings cannot be ascribed to standard chaotic evolution. Moreover, we identify novel short- and mid-range correlations in the Collatz orbits. The Collatz map is hence a paradigmatic example of an arithmetic dynamical system which could also be regarded as displaying key characteristics of an arithmetic statistical physics system, explaining its dynamical richness.

中文翻译：

---

算术动力系统的类随机特征：Collat​​z 冰雹序列

由 Collat​​z 图生成的数值冰雹序列或轨道已被公开，以呈现通常与复杂系统相关的相关特征。尽管算术动力系统迭代规则极其简单，但情况确实如此。事实上，对于一个正整数n, 科拉茨图F读 F（n）=n/2 （ F（n）=3n+1 ） 为了n偶（奇）。为了阐明这个令人惊讶的事实，在这里，我们通过考虑通常用于时间序列的方法，如密码学测试、功率谱、去趋势波动、自相关和熵度量，揭示了 Collat​​z 轨道集合的类随机行为的独特特征。除了证实之前的预测，即科拉茨轨道显示出几何布朗运动的一些全局特性之外，我们的结果同样能够至少启发性地解释其原因。特别是，我们通过综合分析表明，我们的发现不能归因于标准混沌演化。此外，我们还发现了 Collat​​z 轨道中新的短程和中程相关性。因此，Collat​​z 图是算术动力学系统的典型示例，也可以被视为显示算术统计物理系统的关键特征，解释其动力学丰富性。