On the Periodic Orbits of the Dual Logarithmic Derivative Operator

Xiaohang Yu; William Knottenbelt

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arXiv · 2511.21283

On the Periodic Orbits of the Dual Logarithmic Derivative Operator

Xiaohang Yu·William Knottenbelt

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Abstract

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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Xiaohang Yu, William Knottenbelt. 2025-11-26. On the Periodic Orbits of the Dual Logarithmic Derivative Operator. https://arxiv.org/abs/2511.21283

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