揭示对数导数算子的周期轨道结构，给出精确解形式。
On the Periodic Orbits of the Dual Logarithmic Derivative Operator
- 研究复解析框架下对数导数算子的周期行为
- 发现所有非退化2周期解为特定有理函数对，且存在显式例子
- 揭示逻辑斯蒂型函数经变量变换后进入2周期轨道
我们研究复解析设定下双对数导数算子 $\mathcal{A}[f]=\mathrm{d}\ln f/\mathrm{d}\ln x$ 的周期性。证明 $\mathcal{A}$ 存在真正的非退化2周期轨道，并给出一个标准显式例子。由此获得所有非退化2周期解的完整分类，其形式为 $(c a x^{c}/(1-ax^{c}),\, c/(1-ax^{c}))$，其中 $ac\neq 0$。进一步分类了 $\mathcal{A}$ 的所有不动点，表明满足 $\mathcal{A}[f]=f$ 的解均为 $f(x)=1/(a-\ln x)$。作为例证，经对数变量变换后，逻辑斯蒂型函数在一次迭代内即进入2周期族。这些结果给出了算子低周期结构的显式描述，并提供了一个函数空间上算子诱导动力学的可处理范例。
原文摘要 · Abstract (English)
We study the periodic behaviour of the dual logarithmic derivative operator $\mathcal{A}[f]=\mathrm{d}\ln f/\mathrm{d}\ln x$ in a complex analytic setting. We show that $\mathcal{A}$ admits genuinely nondegenerate period-$2$ orbits and identify a canonical explicit example. Motivated by this, we obtain a complete classification of all nondegenerate period-$2$ solutions, which are precisely the rational pairs $(c a x^{c}/(1-ax^{c}),\, c/(1-ax^{c}))$ with $ac\neq 0$. We further classify all fixed points of $\mathcal{A}$, showing that every solution of $\mathcal{A}[f]=f$ has the form $f(x)=1/(a-\ln x)$. As an illustration, logistic-type functions become pre-periodic under $\mathcal{A}$ after a logarithmic change of variables, entering the period-$2$ family in one iterate. These results give an explicit description of the low-period structure of $\mathcal{A}$ and provide a tractable example of operator-induced dynamics on function spaces.
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