On the Periodic Orbits of the Dual Logarithmic Derivative Operator
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914171801567232 |
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| author | Yu, Xiaohang Knottenbelt, William |
| author_facet | Yu, Xiaohang Knottenbelt, William |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_21283 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Periodic Orbits of the Dual Logarithmic Derivative Operator Yu, Xiaohang Knottenbelt, William Dynamical Systems Machine Learning 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. |
| title | On the Periodic Orbits of the Dual Logarithmic Derivative Operator |
| topic | Dynamical Systems Machine Learning |
| url | https://arxiv.org/abs/2511.21283 |