The Discrete Markus-Yamabe Problem for Symmetric Planar Polynomial Maps

TL;DR

This paper investigates the Discrete Markus-Yamabe Problem for symmetric polynomial maps in the plane, characterizing those with Z2 symmetry that satisfy the problem's conditions and developing tools to analyze their spectra.

Contribution

It provides a complete characterization of symmetric polynomial maps satisfying the Discrete Markus-Yamabe Question, focusing on Z2 symmetry, and introduces new spectral analysis tools.

Findings

Only Z2 symmetric nonlinear polynomial maps satisfy the problem.

Normal forms for affirmative solutions are characterized.

New spectral tools for planar polynomial maps are established.

Abstract

We probe deeper into the Discrete Markus-Yamabe Question for polynomial planar maps and into the normal form for those maps which answer this question in the affirmative. Furthermore, in a symmetric context, we show that the only nonlinear equivariant polynomial maps providing an affirmative answer to the Discrete Markus-Yamabe Question are those possessing Z2 as their group of symmetries. We use this to establish two new tools which give information about the spectrum of a planar polynomial map.