Normal families and fixed points of iterates

TL;DR

This paper establishes that a family of holomorphic functions is normal if their second iterates lack fixed points with multipliers exceeding a certain threshold, linking fixed point behavior to normality.

Contribution

It introduces a new criterion for normality based on the multipliers of second iterates of holomorphic functions, connecting fixed point properties to function family behavior.

Findings

Family of holomorphic functions is normal if second iterates lack fixed points with large multipliers.

Results are derived from properties of multipliers of iterated polynomials.

Provides a new perspective on normality criteria in complex dynamics.

Abstract

Let F be a family of holomorphic functions and let K be a constant less than 4. Suppose that for all f in F the second iterate of f does not have fixed points for which the modulus of the multiplier is greater than K. We show that then F is normal. This is deduced from a result about the multipliers of iterated polynomials.