The geometry of the critically-periodic curves in the space of cubic polynomials

TL;DR

This paper presents an algorithm to compute the Euler characteristic of critically periodic curves in cubic polynomial space, with results for periods up to 26, advancing understanding of polynomial dynamics.

Contribution

It introduces a novel algorithm for calculating Euler characteristics of specific polynomial parameter space curves, extending previous theoretical work.

Findings

Successfully computed Euler characteristics for periods up to 26

Demonstrated the algorithm's effectiveness on complex polynomial families

Provided new insights into the topology of critically periodic curves

Abstract

We provide an algorithm for computing the Euler characteristic of the curves $S_p$ in the space of cubic polynomials, consisting of all polynomials with a periodic critical point of period $p$. The curves were introduced in [Milnor, Bonifant-Kiwi-Milnor], and the algorithm applies the main results of [DeMarco-Pilgrim]. The output is shown for periods $p \leq 26$.