AI 中文总结
研究考拉兹动力学中奇偶序列组合结构，通过关联二进制字及函数\(C(d)\)等，在固定长度和密度二进制字集上解决离散优化问题，证明克里斯托费尔词是\(C_{\min}(d)\)唯一最大值，得出相关限制和界，揭示其与组合学联系及结构刚性。
AI 中文摘要
我们研究与加速考拉兹映射相关的奇偶序列的组合结构，旨在识别极值配置并将其与周期轨道的存在联系起来。对于轨道的每个有限序列，我们关联一个二进制字，其中的1编码奇数迭代，并引入一个函数\(C(d)\)，它为迭代提供显式表达式并表征可能的周期循环。我们定义了一个与周期轨道的循环结构兼容的二进制字上的自然旋转作用，并将函数\(C_{\min}(d)\)视为每个旋转类的规范代表。在此设置下，我们在固定长度和规定密度的二进制字集合上制定并解决了一个离散优化问题。我们证明，在长度为\(N\)且恰好有\(r\)个1的二进制字集合\(D_{N,r}\)上，克里斯托费尔词在旋转下是\(C_{\min}(d)\)的唯一最大值，从而在考拉兹问题的动力学与平衡字的经典理论之间建立了直接联系。因此，我们得到了关于非平凡循环可能存在性的限制，并根据轨道长度和奇数迭代比例得出了轨道最小元素的显式界。这些结果表明，奇偶序列的组合结构对考拉兹动力学施加了强大的约束，并表明极值配置由字组合学中的经典对象控制，表现出明显的结构刚性。
英文摘要
We study the combinatorial structure of parity sequences associated with the accelerated Collatz map with the goal of identifying extremal configurations and relating them to the existence of periodic orbits. To each finite sequence of an orbit, we associate a binary word whose ones encode the odd iterates, and we introduce a functional $C(d)$ on such words which provides an explicit expression for the iterates and characterizes possible periodic cycles. We define a natural rotation action on binary words, compatible with the cyclic structure of periodic orbits, and consider the functional $C_{\min}(d)$ as a canonical representative of each rotation class. In this setting, we formulate and solve a discrete optimization problem on the set of binary words of fixed length and prescribed density. We prove that Christoffel words are, up to rotation, the unique maximizers of $C_{\min}(d)$ on $D_{N,r}$, the set of binary words of length $N$ with exactly $r$ ones, thereby establishing a direct connection between the dynamics of the Collatz problem and the classical theory of balanced words. As a consequence, we obtain restrictions on the possible existence of nontrivial cycles and derive explicit bounds for the minimum element of an orbit in terms of its length and the proportion of odd iterates. These results show that the combinatorial structure of parity sequences imposes strong constraints on Collatz dynamics and suggest that extremal configurations are governed by classical objects from the combinatorics on words, exhibiting a pronounced structural rigidity.
Comments18 pages, 1 table
