{"ID":2836359,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2511.21283","arxiv_id":"2511.21283","title":"On the Periodic Orbits of the Dual Logarithmic Derivative Operator","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.","short_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 class...","url_abs":"https://arxiv.org/abs/2511.21283","url_pdf":"https://arxiv.org/pdf/2511.21283v1","authors":"[\"Xiaohang Yu\",\"William Knottenbelt\"]","published":"2025-11-26T11:14:32Z","proceeding":"math.DS","tasks":"[\"math.DS\",\"cs.LG\"]","methods":"[]","has_code":false}