Cubic polynomials with periodic cycles of a specified multiplier

TL;DR

This paper studies cubic polynomials with a marked periodic point of a given period and multiplier, revealing infinite cases for period 1, finiteness for periods greater than 2, and detailed structure for period 2.

Contribution

It characterizes the structure of cubic polynomials with specified periodic points and multipliers over function fields, especially describing the case for period 2.

Findings

Infinite such polynomials for period N=1

Finite for N>2

Complete description for N=2 over certain field extensions

Abstract

We consider cubic polynomials f(z)=z^3+az+b defined over the function field C(L), with a marked point of period N and multiplier L. In the case N=1, there are infinitely many such objects, and in the case N>2, only finitely many. The case N=2 has particularly rich structure, and we are able to describe all such cubic polynomials defined over the field obtained by adjoining to C the mth roots of L, for all L.