Julia sets of Newton maps of real quadratic polynomial maps on the plane

TL;DR

This paper numerically investigates the dynamics of Newton maps for simple polynomial transformations on the plane, focusing on their limit sets and confirming previous conjectures about their behavior.

Contribution

It provides the first detailed numerical analysis of Newton maps for quadratic polynomial maps in the plane, supporting existing conjectures about their dynamics.

Findings

Numerical evidence supports conjectures on the limit sets of Newton maps.

Analysis covers maps with linear and quadratic components.

Results enhance understanding of polynomial map dynamics in the plane.

Abstract

We study numerically the $\alpha$- and $\omega$-limits of the Newton maps of two of the most elementary families of polynomial transformations on the plane: those with a linear component and those with both components of degree two. Our results are fully consistent with some conjectures we posed in a recent work about the dynamics of Newton maps.