On the Periodic Orbits of the Dual Logarithmic Derivative Operator

2025/11/26 by X.M. Yu, Yu, Xiaohang, W.J. Knottenbelt +1
Mathematics · #Holomorphic and Operator Theory #Advanced Differential Equations and Dynamical Systems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2511.21283

Abstract

We study the periodic behaviour of the dual logarithmic derivative operator A[f]=dln f/dln x in a complex analytic setting. We show that 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≠ 0. We further classify all fixed points of A, showing that every solution of A[f]=f has the form f(x)=1/(a-ln x). As an illustration, logistic-type functions become pre-periodic under 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 A and provide a tractable example of operator-induced dynamics on function spaces.

Citations

• Logarithmic Derivatives and Generalized Dynkin Operators

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