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On the Periodic Orbits of the Dual Logarithmic Derivative Operator

Xiaohang Yu, William Knottenbelt

TL;DR

The paper analyzes the iterative dynamics of the dual logarithmic derivative operator $\\mathcal{A}[f] = \\frac{d\\ln f}{d\\ln x}$ on domains in $\\mathbb{C}$, establishing explicit low-period structure in the complex analytic setting. It proves that genuinely nondegenerate period-2 orbits exist and provides a canonical example, then completely classifies all nondegenerate period-2 orbits as the two-parameter rational family $f_1(x)=\\frac{c a x^{c}}{1-a x^{c}}$, $f_2(x)=\\frac{c}{1-a x^{c}}$, along with a full description of fixed points $f(x)=\\frac{1}{a-\\ln x}$. A logistic-type pre-periodic phenomenon under a logarithmic change of variables demonstrates how generic initial data can flow into the classified period-2 family after finitely many iterations. These results yield an explicit, tractable instance of operator-induced dynamics on function spaces, highlighting a transparent low-period structure and a concrete pre-periodic mechanism with potential applications to growth and scaling analyses.

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.

On the Periodic Orbits of the Dual Logarithmic Derivative Operator

TL;DR

The paper analyzes the iterative dynamics of the dual logarithmic derivative operator on domains in , establishing explicit low-period structure in the complex analytic setting. It proves that genuinely nondegenerate period-2 orbits exist and provides a canonical example, then completely classifies all nondegenerate period-2 orbits as the two-parameter rational family , , along with a full description of fixed points . A logistic-type pre-periodic phenomenon under a logarithmic change of variables demonstrates how generic initial data can flow into the classified period-2 family after finitely many iterations. These results yield an explicit, tractable instance of operator-induced dynamics on function spaces, highlighting a transparent low-period structure and a concrete pre-periodic mechanism with potential applications to growth and scaling analyses.

Abstract

We study the periodic behaviour of the dual logarithmic derivative operator in a complex analytic setting. We show that admits genuinely nondegenerate period- orbits and identify a canonical explicit example. Motivated by this, we obtain a complete classification of all nondegenerate period- solutions, which are precisely the rational pairs with . We further classify all fixed points of , showing that every solution of has the form . As an illustration, logistic-type functions become pre-periodic under after a logarithmic change of variables, entering the period- family in one iterate. These results give an explicit description of the low-period structure of and provide a tractable example of operator-induced dynamics on function spaces.

Paper Structure

This paper contains 24 sections, 6 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Dual Logarithmic Derivatives.
  3. Function-space Fixed-point Theory.
  4. Operator Dynamics.
  5. Our Contributions.
  6. Definitions
  7. The Operator $\mathcal{A}$
  8. Period-2 Orbit
  9. Fixed Point Function
  10. A Canonical Period-2 Orbit
  11. Classification of Periodic-2 Orbits
  12. Period-2 Forces Constant Difference
  13. Complete Classification of Periodic-2 Orbits
  14. Classification of Fixed Points
  15. Constant Functions Are Not Fixed Points
  16. ...and 9 more sections

Key Result

Proposition 1

Fix $a\in\mathbb{C}\setminus\{0\}$ and define for all $x$ in any domain $U\subset\mathbb{C}$ on which $1-a x\neq 0$. Then $(f_1,f_2)$ forms a nondegenerate period-$2$ orbit of $\mathcal{A}$.

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Lemma 2
  • ...and 5 more