On the Periodic Orbits of the Dual Logarithmic Derivative Operator

TL;DR

This paper investigates the periodic behavior of the dual logarithmic derivative operator in complex analysis, classifies its period-2 solutions and fixed points, and explores the dynamics induced on function spaces.

Contribution

It provides a complete classification of nondegenerate period-2 solutions and fixed points of the operator, offering explicit forms and understanding of its low-period dynamics.

Findings

Existence of nondegenerate period-2 orbits

Complete classification of period-2 solutions as rational pairs

Explicit description of fixed points as functions of the form 1/(a−ln x)

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.