Perturbations of weakly expanding critical orbits

TL;DR

This paper investigates how small changes in certain complex functions with summable critical points affect the behavior of their critical orbits, revealing a structured relationship between parameter and dynamical derivatives.

Contribution

It establishes the existence of an r-dimensional manifold around such functions where parameter and dynamical derivatives are proportionally related along critical orbits.

Findings

Existence of an r-dimensional manifold near functions with summable critical points.

Non-zero limiting ratio between parameter and dynamical derivatives along critical orbits.

Structured relationship between parameter changes and orbit dynamics.

Abstract

Let f be a polynomial or a rational function which has r summable critical points. We prove that there exists an r-dimensional manifold in an appropriate space containing f such that for every smooth curve in it through f, the ratio between parameter and dynamical derivatives along forward iterates of at least one these summable points tends to a non-zero number.