On the Periodic Orbits of the Dual Logarithmic Derivative Operator

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 xc/(1-axc),, c/(1-axc)) with acneq 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.