发表机构
School of Computing and Data Sciences, FLAME University; Allos AI Limited（FLAME大学计算与数据科学学院; Allos AI有限公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究平移Alladi-Erdős整数映射的动力学，通过探索参数空间发现吸引子周期的分岔结构，并利用初等数论解释分岔图，揭示简单映射产生的丰富算术动力学。
AI 中文摘要
整数映射是定义在自然数上的离散动力系统。本文研究了平移Alladi-Erdős映射的动力学，这是一族单参数整数映射，其中每个合数被映射到其素因子之和，而每个素数被映射到$n+A$，其中$A \in \mathbb{N}$是固定的平移参数。通过系统地探索$2 \leq A \leq 10^5$的参数空间，我们揭示了丰富的分岔结构，其特征是随着平移参数的变化，吸引子周期出现、消失和重组。我们发现，尽管对于给定的$A$值，多个吸引子可能共存，但对于大多数$A$值，几乎所有自然数都属于仅两个主要吸引子的吸引域。利用初等数论论证，我们解释了观察到的分岔图。我们进一步刻画了吸引子周期，并量化了其吸引域大小在参数空间中的分布。我们的结果揭示了由一个极其简单的整数映射产生的算术动力学中出乎意料的丰富图景。
英文摘要
Integer maps are discrete dynamical systems defined on the natural numbers. In this paper, we investigate the dynamics of the shifted Alladi-Erd{\H o}s map, a one-parameter family of integer maps in which each composite number is mapped to the sum of its prime factors, while each prime is mapped to $n+A$, where $A \in \mathbb{N}$ is a fixed shift parameter. By systematically exploring the parameter space for $2 \leq A \leq 10^5$, we uncover a rich bifurcation structure characterized by the emergence, disappearance, and reorganization of attractor cycles as the shift parameter varies. We find that although several attractors may coexist for a given value of $A$, for most values of $A$, almost all natural numbers belong to the basins of only two dominant attractors. Using elementary number theoretic arguments, we explain the observed bifurcation diagram. We further characterize the attractor cycles and quantify the distribution of their basin sizes across the parameter space. Our results reveal an unexpectedly rich landscape of arithmetic dynamics arising from a remarkably simple integer map. %and provide a systematic characterization of how attractors and their basins evolve as shift parameter is varied.
Comments7 pages, 10 figures
